[ibrwpoia] full coverage by uniform sampling

B different balls in an urn.  uniformly draw with replacement N times.  what is the probability that all B balls are seen during the N draws?

given B, for what value of N is the probability (about) 50%?

simulation in Pari/GP:

u(trials, B, N)= my(success=0); for(c=1, trials, my(a=vector(B)); for(i=1, N, r=1+random(B); a[r]=1); good=1; for(i=1, B, if(0==a[i], good=0; break)); success+=good); 1.0*success/trials;

B N
2 2
4 7
8 19-20
10 26-27
16 50
20 67
30 113
32 122
40 162
50 213-214
60 267-268
64 289-290

B=64 balls, N=290 draws:
u(10^5, 64, 290)
0.50303

cpu time = 21,920 ms, real time = 21,924 ms.

exact answer: 1 - probability of failure by the inclusion-exclusion principle, simplifies to:

v(B, N)= sum(i=0, B, (-1)^i * (B-i)^N * binomial(B, i)) /B^N

1.*v(64,290)
0.504257282

exact bounds:

10 [26, 27]
20 [66, 67]
30 [112, 113]
40 [161, 162]
50 [213, 214]
60 [267, 268]
70 [322, 323]
80 [379, 380]
90 [437, 438]
100 [496, 497]
110 [556, 557]
120 [618, 619]
130 [679, 680]
140 [742, 743]
150 [806, 807]
160 [870, 871]
170 [934, 935]
180 [1000, 1001]
190 [1066, 1067]
200 [1132, 1133]
210 [1199, 1200]
220 [1266, 1267]
230 [1334, 1335]
240 [1402, 1403]
250 [1471, 1472]
260 [1540, 1541]

2 [2, 2]
4 [6, 7]
8 [19, 20]
16 [50, 51]
32 [122, 123]
64 [289, 290]
128 [667, 668]
256 [1512, 1513]
512 [3381, 3382]
1024 [7472, 7473]
2048 [16364, 16365]
4096 [35569, 35570]

the growth rate of N seems to be O(B*log B).  let's use N = B*log B exactly and calculate the probability of full coverage:

v(B, round(B*log(B)))

B N v
4 6 0.380859375
8 17 0.365562211
16 44 0.347014424
32 111 0.363466498
64 266 0.364065310
128 621 0.366295109
256 1420 0.367804277
512 3194 0.367507854
1024 7098 0.367763404
2048 15615 0.367750482
4096 34070 0.367873303

the probability seems to be converging to exp(-1) = 0.367879441 .

how many draws N are necessary to achieve at least a given probability of full coverage?

previously similar: Euler's Game of Coincidence

[zxvpfzdb] decade primes and gaps

4 consecutive primes as near to each other as they can possibly be is called a prime 4-tuplet or prime quadruplet or prime decade because, for those starting greater than 10, the last digits (the one's place, unit's place, least significant digit) have the pattern "1 3 7 9"; that is, they all fall in one decade.  it is never "3 7 9 1" nor "7 9 1 3" nor "9 1 3 7".

prime gaps of length 10 are aesthetically the dual.

inspired by memorizing primes in order.

below are decades and gaps up to 3011.

the first three prime quadruplets are the weird ones, not having last digits "1 3 7 9".

97 101 103 107 109 113 is a prime 6-tuplet containing two prime 5-tuplets and (only) one prime 4-tuplet.  the next prime 5-tuplet is 1481 1483 1487 1489 1493.  the next prime 6-tuplet is 16057 16061 16063 16067 16069 16073.  prime decades generally must be separated by a multiple of 30 (future post bqazxnke).

we give the gap size only if greater than 10.

decade 2 3 5 7
decade 3 5 7 11
decade 5 7 11 13
decade 11 13 17 19
decade 101 103 107 109
gap 113 127 size 14
gap 139 149
gap 181 191
decade 191 193 197 199
gap 199 211 size 12
gap 211 223 size 12
gap 241 251
gap 283 293
gap 293 307 size 14
gap 317 331 size 14
gap 337 347
gap 409 419
gap 421 431
gap 467 479 size 12
gap 509 521 size 12
gap 523 541 size 18
gap 547 557
gap 577 587
gap 619 631 size 12
gap 631 641
gap 661 673 size 12
gap 691 701
gap 709 719
gap 773 787 size 14
gap 787 797
gap 797 809 size 12
gap 811 821
decade 821 823 827 829
gap 829 839
gap 839 853 size 14
gap 863 877 size 14
gap 887 907 size 20
gap 919 929
gap 953 967 size 14
gap 997 1009 size 12
gap 1021 1031
gap 1039 1049
gap 1051 1061
gap 1069 1087 size 18
gap 1129 1151 size 22
gap 1153 1163
gap 1171 1181
gap 1201 1213 size 12
gap 1237 1249 size 12
gap 1249 1259
gap 1259 1277 size 18
gap 1307 1319 size 12
gap 1327 1361 size 34
gap 1381 1399 size 18
gap 1399 1409
gap 1409 1423 size 14
gap 1459 1471 size 12
gap 1471 1481
decade 1481 1483 1487 1489
gap 1499 1511 size 12
gap 1511 1523 size 12
gap 1531 1543 size 12
gap 1583 1597 size 14
gap 1627 1637
gap 1637 1657 size 20
gap 1669 1693 size 24
gap 1699 1709
gap 1709 1721 size 12
gap 1723 1733
gap 1759 1777 size 18
gap 1789 1801 size 12
gap 1801 1811
gap 1811 1823 size 12
gap 1831 1847 size 16
gap 1847 1861 size 14
decade 1871 1873 1877 1879
gap 1879 1889
gap 1889 1901 size 12
gap 1913 1931 size 18
gap 1933 1949 size 16
gap 1951 1973 size 22
gap 2017 2027
gap 2029 2039
gap 2039 2053 size 14
gap 2053 2063
gap 2069 2081 size 12
decade 2081 2083 2087 2089
gap 2089 2099
gap 2099 2111 size 12
gap 2113 2129 size 16
gap 2143 2153
gap 2161 2179 size 18
gap 2179 2203 size 24
gap 2221 2237 size 16
gap 2251 2267 size 16
gap 2297 2309 size 12
gap 2311 2333 size 22
gap 2357 2371 size 14
gap 2399 2411 size 12
gap 2423 2437 size 14
gap 2447 2459 size 12
gap 2477 2503 size 26
gap 2503 2521 size 18
gap 2521 2531
gap 2557 2579 size 22
gap 2579 2591 size 12
gap 2593 2609 size 16
gap 2621 2633 size 12
gap 2633 2647 size 14
gap 2647 2657
gap 2719 2729
gap 2731 2741
gap 2753 2767 size 14
gap 2767 2777
gap 2777 2789 size 12
gap 2803 2819 size 16
gap 2819 2833 size 14
gap 2861 2879 size 18
gap 2887 2897
gap 2917 2927
gap 2927 2939 size 12
gap 2939 2953 size 14
gap 2971 2999 size 28
gap 3001 3011

gaps become increasingly common as primes thin out, so we stop listing them.  here are just decades from 3000 to a million:

decade 3251 3253 3257 3259
decade 3461 3463 3467 3469
decade 5651 5653 5657 5659
decade 9431 9433 9437 9439
decade 13001 13003 13007 13009
decade 15641 15643 15647 15649
decade 15731 15733 15737 15739
decade 16061 16063 16067 16069
decade 18041 18043 18047 18049
decade 18911 18913 18917 18919
decade 19421 19423 19427 19429
decade 21011 21013 21017 21019
decade 22271 22273 22277 22279
decade 25301 25303 25307 25309
decade 31721 31723 31727 31729
decade 34841 34843 34847 34849
decade 43781 43783 43787 43789
decade 51341 51343 51347 51349
decade 55331 55333 55337 55339
decade 62981 62983 62987 62989
decade 67211 67213 67217 67219
decade 69491 69493 69497 69499
decade 72221 72223 72227 72229
decade 77261 77263 77267 77269
decade 79691 79693 79697 79699
decade 81041 81043 81047 81049
decade 82721 82723 82727 82729
decade 88811 88813 88817 88819
decade 97841 97843 97847 97849
decade 99131 99133 99137 99139
decade 101111 101113 101117 101119
decade 109841 109843 109847 109849
decade 116531 116533 116537 116539
decade 119291 119293 119297 119299
decade 122201 122203 122207 122209
decade 135461 135463 135467 135469
decade 144161 144163 144167 144169
decade 157271 157273 157277 157279
decade 165701 165703 165707 165709
decade 166841 166843 166847 166849
decade 171161 171163 171167 171169
decade 187631 187633 187637 187639
decade 194861 194863 194867 194869
decade 195731 195733 195737 195739
decade 201491 201493 201497 201499
decade 201821 201823 201827 201829
decade 217361 217363 217367 217369
decade 225341 225343 225347 225349
decade 240041 240043 240047 240049
decade 243701 243703 243707 243709
decade 247601 247603 247607 247609
decade 247991 247993 247997 247999
decade 257861 257863 257867 257869
decade 260411 260413 260417 260419
decade 266681 266683 266687 266689
decade 268811 268813 268817 268819
decade 276041 276043 276047 276049
decade 284741 284743 284747 284749
decade 285281 285283 285287 285289
decade 294311 294313 294317 294319
decade 295871 295873 295877 295879
decade 299471 299473 299477 299479
decade 300491 300493 300497 300499
decade 301991 301993 301997 301999
decade 326141 326143 326147 326149
decade 334421 334423 334427 334429
decade 340931 340933 340937 340939
decade 346391 346393 346397 346399
decade 347981 347983 347987 347989
decade 354251 354253 354257 354259
decade 358901 358903 358907 358909
decade 361211 361213 361217 361219
decade 375251 375253 375257 375259
decade 388691 388693 388697 388699
decade 389561 389563 389567 389569
decade 392261 392263 392267 392269
decade 394811 394813 394817 394819
decade 397541 397543 397547 397549
decade 397751 397753 397757 397759
decade 402131 402133 402137 402139
decade 402761 402763 402767 402769
decade 412031 412033 412037 412039
decade 419051 419053 419057 419059
decade 420851 420853 420857 420859
decade 427241 427243 427247 427249
decade 442571 442573 442577 442579
decade 444341 444343 444347 444349
decade 452531 452533 452537 452539
decade 463451 463453 463457 463459
decade 465161 465163 465167 465169
decade 467471 467473 467477 467479
decade 470081 470083 470087 470089
decade 477011 477013 477017 477019
decade 490571 490573 490577 490579
decade 495611 495613 495617 495619
decade 500231 500233 500237 500239
decade 510611 510613 510617 510619
decade 518801 518803 518807 518809
decade 536441 536443 536447 536449
decade 536771 536773 536777 536779
decade 539501 539503 539507 539509
decade 549161 549163 549167 549169
decade 559211 559213 559217 559219
decade 563411 563413 563417 563419
decade 570041 570043 570047 570049
decade 572651 572653 572657 572659
decade 585911 585913 585917 585919
decade 594821 594823 594827 594829
decade 597671 597673 597677 597679
decade 607301 607303 607307 607309
decade 622241 622243 622247 622249
decade 626621 626623 626627 626629
decade 632081 632083 632087 632089
decade 632321 632323 632327 632329
decade 633461 633463 633467 633469
decade 633791 633793 633797 633799
decade 654161 654163 654167 654169
decade 657491 657493 657497 657499
decade 661091 661093 661097 661099
decade 663581 663583 663587 663589
decade 664661 664663 664667 664669
decade 666431 666433 666437 666439
decade 680291 680293 680297 680299
decade 681251 681253 681257 681259
decade 691721 691723 691727 691729
decade 705161 705163 705167 705169
decade 715151 715153 715157 715159
decade 734471 734473 734477 734479
decade 736361 736363 736367 736369
decade 739391 739393 739397 739399
decade 768191 768193 768197 768199
decade 773021 773023 773027 773029
decade 795791 795793 795797 795799
decade 803441 803443 803447 803449
decade 814061 814063 814067 814069
decade 822161 822163 822167 822169
decade 823721 823723 823727 823729
decade 829721 829723 829727 829729
decade 833711 833713 833717 833719
decade 837071 837073 837077 837079
decade 845981 845983 845987 845989
decade 854921 854923 854927 854929
decade 855731 855733 855737 855739
decade 857951 857953 857957 857959
decade 875261 875263 875267 875269
decade 876011 876013 876017 876019
decade 881471 881473 881477 881479
decade 889871 889873 889877 889879
decade 907391 907393 907397 907399
decade 930071 930073 930077 930079
decade 938051 938053 938057 938059
decade 946661 946663 946667 946669
decade 954971 954973 954977 954979
decade 958541 958543 958547 958549
decade 959471 959473 959477 959479
decade 976301 976303 976307 976309
decade 978071 978073 978077 978079
decade 983441 983443 983447 983449

prime decades can only occur in decades 10*(3*n + 1) + {1,3,7,9}.  here are "composite decades" when all numbers in a candidate prime decade are composite:

1331 = 11^3
1333 = 31 * 43
1337 = 7 * 191
1339 = 13 * 103

1961 = 37 * 53
1963 = 13 * 151
1967 = 7 * 281
1969 = 11 * 179

2321 = 11 * 211
2323 = 23 * 101
2327 = 13 * 179
2329 = 17 * 137

2561 = 13 * 197
2563 = 11 * 233
2567 = 17 * 151
2569 = 7 * 367

2981 = 11 * 271
2983 = 19 * 157
2987 = 29 * 103
2989 = 7^2 * 61

3281 = 17 * 193
3283 = 7^2 * 67
3287 = 19 * 173
3289 = 11 * 13 * 23

3971 = 11 * 19^2
3973 = 29 * 137
3977 = 41 * 97
3979 = 23 * 173

4031 = 29 * 139
4033 = 37 * 109
4037 = 11 * 367
4039 = 7 * 577

4061 = 31 * 131
4063 = 17 * 239
4067 = 7^2 * 83
4069 = 13 * 313

4181 = 37 * 113
4183 = 47 * 89
4187 = 53 * 79
4189 = 59 * 71

4301 = 11 * 17 * 23
4303 = 13 * 331
4307 = 59 * 73
4309 = 31 * 139

4571 = 7 * 653
4573 = 17 * 269
4577 = 23 * 199
4579 = 19 * 241

4841 = 47 * 103
4843 = 29 * 167
4847 = 37 * 131
4849 = 13 * 373

6401 = 37 * 173
6403 = 19 * 337
6407 = 43 * 149
6409 = 13 * 17 * 29

6431 = 59 * 109
6433 = 7 * 919
6437 = 41 * 157
6439 = 47 * 137

6641 = 29 * 229
6643 = 7 * 13 * 73
6647 = 17^2 * 23
6649 = 61 * 109

7091 = 7 * 1013
7093 = 41 * 173
7097 = 47 * 151
7099 = 31 * 229

7271 = 11 * 661
7273 = 7 * 1039
7277 = 19 * 383
7279 = 29 * 251

7421 = 41 * 181
7423 = 13 * 571
7427 = 7 * 1061
7429 = 17 * 19 * 23

8021 = 13 * 617
8023 = 71 * 113
8027 = 23 * 349
8029 = 7 * 31 * 37

8471 = 43 * 197
8473 = 37 * 229
8477 = 7^2 * 173
8479 = 61 * 139

8651 = 41 * 211
8653 = 17 * 509
8657 = 11 * 787
8659 = 7 * 1237

8981 = 7 * 1283
8983 = 13 * 691
8987 = 11 * 19 * 43
8989 = 89 * 101

9071 = 47 * 193
9073 = 43 * 211
9077 = 29 * 313
9079 = 7 * 1297

9701 = 89 * 109
9703 = 31 * 313
9707 = 17 * 571
9709 = 7 * 19 * 73

9911 = 11 * 17 * 53
9913 = 23 * 431
9917 = 47 * 211
9919 = 7 * 13 * 109

two decades in a row can have the same non-empty prime pattern (same last digits) only when one decade is a candidate (but failed) prime decade 10*(3*n + 1).  here are those repeated patterns, with the candidate (but failed) prime decade marked "decade".

373 379 decade
383 389

787
797 decade

1249 decade
1259

1399 decade
1409

1543 1549 decade
1553 1559

1801
1811 decade

2341 2347
2351 2357 decade

2767
2777 decade

2887
2897 decade

2917
2927 decade

3109 decade
3119

3313 3319 decade
3323 3329

4363 decade
4373

4441 4447
4451 4457 decade

4993 4999 decade
5003 5009

5323 decade
5333

5581
5591 decade

6091
6101 decade

6163 decade
6173

6247
6257 decade

6481
6491 decade

6961 6967
6971 6977 decade

7069 decade
7079

7177
7187 decade

7507
7517 decade

8209 decade
8219

8563 decade
8573

8731 8737
8741 8747 decade

8941
8951 decade

9511
9521 decade

9613 9619 decade
9623 9629

9733 9739 decade
9743 9749

Pari/GP code for prime decades:

sz=4; width=8; a=List(vector(sz)); forprime(p=0, 10^5, listpop(a, 1); listput(a, p); if(a[1] && (a[sz]-a[1]<=width), print1("decade");for(i=1, sz, print1(" ",a[i])); print))

to explore prime constellations other than prime quadruplets (sz=4, width=8), set (sz,width) to one of the following: (2,2) (3,6) (4,8) (5,12) (6,16) (7,20) and more terms at OEIS A008407.  (future work jqvjozns)

prime gaps:

old=2; forprime(p=0, 3012, d=p-old; if(d>=10, print1("gap "old" "p); if(d>10, print(" size "d), print())); old=p)

composite decades:

decade=[1,3,7,9]; for(i=0, 100, d=30*i+10; z=1; for(j=1, 4, if(isprime(d+decade[j]), z=0; break)); if(z, print(); for(j=1, 4 ,e=d+decade[j]; print(e" ", factor(e)))))

consecutive decades having the same prime pattern (but not both empty):

decade=[1,3,7,9];
pattern(n)=my(s=0); for(j=0, 3, i=4-j; s*=2; s+=isprime(n+decade[i])); s;
listdecade(n,trail)=for(i=n, n+9, if(isprime(i), print1(" "i))); print(trail);
for(i=0, 400, n=10*(3*i+1); p=pattern(n); if(!p,next); if(pattern(n-10)==p, listdecade(n-10," "); listdecade(n," decade"); print()); if(pattern(n+10)==p, listdecade(n," decade "); listdecade(n+10,""); print()))

[clyaebic] worst chess evar

we set up some common chess openings, direct one side to play the worst moves according to MultiPV of the Stockfish chess engine and the other side to play the best moves according to Stockfish, and watch hilarity ensue.

Stockfish 16-1 (Debian sid; NB: this is Stockfish 16 not 16.1; the -1 is Debian versioning), hash 4096M, depth 24, no multithreading, 5 piece syzygy endgame tablebase. multipv set to 400 to find the worst move; multipv disabled when finding the best move. engine restarted (Hash cleared) between each move.  there are (probably) instances of multiple possible moves with the same (terrible) evaluation score; we choose whichever Stockfish MultiPV chooses to sort as last.

in each game below, the side which gets checkmated is unsurprisingly the side playing worst moves, and the other side is playing best moves.  even though these are all quick checkmates, they are not helpmates: the side playing good moves does not know the other side will be playing bad moves so plays best moves preparing the the best response.

(future work: helpmates after common openings.)

many variations of Fool's Mate, Botez Gambit, and Bongcloud happen.  the computer knows the memes.

no opening moves, white playing worst moves:

1.g4 d5 2.f4 Bxg4 3.e3 Bxd1 4.Bc4 dxc4 5.a3 Qd5 6.Ne2 Qxh1+ 7.Ng1 Qxg1#

the next opening is hilariously classified by Scid as ECO code B00b [Reversed Grob (Borg/Basman Defence)].  no opening moves, white playing best moves, black playing worst moves:

1.e4 g5 2.d4 f5 3.Qh5#

1.e4 {end of scripted opening} e5 2.Ba6 Nxa6 3.g4 d5 4.f4 Qh4+ 5.Ke2 Bxg4+ 6.Ke3 exf4+ 7.Kd4 Qf2+ 8.Kxd5 Qc5#

1.e4 e5 {end of opening} 2.Nf3 Qg5 3.Nxg5 Kd8 4.Nxf7+ Ke8 5.Nxh8 h5 6.Qxh5+ Ke7 7.Qxe5+ Kd8 8.Nf7#

1.e4 e5 2.Nf3 {end of opening} Nc6 3.Nd4 exd4 4.Ba6 bxa6 5.Ke2 Rb8 6.Kd3 Qh4 7.Kc4 d5+ 8.Kd3 Qxe4#

1.e4 e5 2.Nf3 Nc6 {end of opening} 3.Bb5 Qg5 4.Nxg5 Kd8 5.Nxf7+ Ke8 6.Nxh8 h5 7.Qxh5+ g6 8.Qxg6+ Kd8 9.Qxg8 Rb8 10.Qxf8#

1.e4 c5 {end of opening} 2.Ba6 Nxa6 3.g4 d5 4.Ke2 dxe4 5.f4 Bxg4+ 6.Ke3 Qd4#

1.e4 c5 {end of opening} 2.Nf3 c4 3.Bxc4 h6 4.Bxf7+ Kxf7 5.Ne5+ Ke8 6.Qh5+ g6 7.Qxg6#

1.e4 e6 {end of opening} 2.Ba6 Nxa6 3.b4 Nxb4 4.Qg4 Nxc2+ 5.Ke2 Nxa1 6.Qg5 Qxg5 7.Kd3 Qb5+ 8.Kd4 Bc5+ 9.Ke5 Be3#

1.e4 e6 {end of opening} 2.d4 Qg5 3.Bxg5 Ba3 4.Nxa3 c5 5.Nb5 g6 6.Nd6+ Kf8 7.Qf3 Kg7 8.Qxf7#

1.e4 e6 2.d4 {end of opening} d5 3.Bh6 Nxh6 4.Qg4 Nxg4 5.Kd1 Nxf2+ 6.Ke1 Nxh1 7.h4 Qxh4+ 8.Ke2 Qf2+ 9.Kd3 Qe1 10.b4 Nf2#

1.e4 c6 {end of opening} 2.Ba6 Nxa6 3.Qg4 d5 4.c4 Bxg4 5.f4 Nb4 6.h3 Nd3+ 7.Kf1 Qb6 8.g3 Qf2#

1.e4 c6 {end of opening} 2.d4 g5 3.Bxg5 f5 4.Qh5#

1.e4 c6 2.d4 {end of opening} d5 3.Qg4 Bxg4 4.Bh6 Nxh6 5.Kd2 dxe4 6.c4 Qxd4+ 7.Ke1 Qd1#

1.e4 d6 {end of opening} 2.Qg4 Bxg4 3.Ba6 Nxa6 4.c4 Nb4 5.e5 dxe5 6.Kf1 Qd3+ 7.Ke1 Nc2#

1.e4 d6 {end of opening} 2.d4 Bh3 3.Nxh3 f5 4.exf5 Kf7 5.Qh5+ Kf6 6.Bg5+ Kxf5 7.Qf7+ Kg4 8.f3#

1.e4 d6 2.d4 {end of opening} Nf6 3.Qg4 Bxg4 4.Ba6 Nxa6 5.Bh6 gxh6 6.b4 Nxb4 7.Kd2 Nxe4+ 8.Ke1 Nxc2+ 9.Kf1 Qd7 10.Nd2 Nxd2#

1.e4 d6 2.d4 Nf6 {end of opening} 3.Nc3 Bh3 4.Nxh3 Nd5 5.Nxd5 f5 6.exf5 Nd7 7.Qh5+ g6 8.fxg6 Rc8 9.gxh7#

1.e4 g6 {end of opening} 2.Qh5 gxh5 3.Ba6 Nxa6 4.c3 Nc5 5.a3 Nd3+ 6.Kd1 d5 7.exd5 Qxd5 8.Ne2 Qb3#

1.e4 g6 {end of opening} 2.d4 g5 3.Bxg5 f5 4.Qh5#

1.e4 g6 2.d4 {end of opening} Bg7 3.Qh5 gxh5 4.Bh6 Nxh6 5.d5 Bxb2 6.Nc3 Bxc3+ 7.Ke2 Bxa1 8.Kd3 c6 9.Kc4 Qa5 10.c3 Qxc3#

1.d4 {end of opening} g5 2.Bxg5 e6 3.Bxd8 Nf6 4.Bxf6 Ba3 5.Nxa3 O-O 6.e3 e5 7.Qg4#

Trompowsky but worse:

1.d4 {end of opening} Nf6 2.Bh6 gxh6 3.Kd2 c5 4.Kc3 Ne4+ 5.Kb3 Qb6+ 6.Ka3 Qb4#

1.d4 Nf6 {end of opening} 2.c4 Nd5 3.cxd5 Nc6 4.dxc6 f5 5.Nf3 Kf7 6.cxd7 Kg8 7.Qb3+ e6 8.Qxe6#

1.d4 d5 {end of opening} 2.Bh6 Nxh6 3.Kd2 c5 4.Kc3 Qa5+ 5.Kb3 c4#

1.d4 d5 {end of opening} 2.c4 Bh3 3.Nxh3 Kd7 4.cxd5 Nc6 5.dxc6+ Ke6 6.Ng5+ Kf5 7.Qd3+ Kg4 8.Qh3#

1.Nf3 {end of opening} g5 2.Nxg5 f6 3.e4 f5 4.Qh5#

1.c4 {end of opening} g5 2.d4 f5 3.Bxg5 e6 4.Bxd8 Bc5 5.dxc5 a6 6.Qd4 Nh6 7.Qxh8+ Ng8 8.Qxg8#

challenge: start from a reasonable position and maximize the number of consecutive worst moves before checkmate.

[wyhgtulu] nearby prime differences

start with a set of consecutive primes.  compute all pairs positive differences among the set, and count the number of times each difference occurs.

note that we are counting differences between primes not necessarily consecutive.  (in contrast, prime gaps, another well studied topic, are differences between consecutive primes.)

running through the differences of all pairs seems like it could be done cleverly faster than quadratic time, but I could not think of a way.  doing all pairs of sets of a million numbers took about half an hour in C++.

among the first 10 odd primes (3 5 7 11 13 17 19 23 29 31), below are the differences from most frequent to least frequent.  each parenthesized record has the format
(difference number-of-occurrences #rank)
where rank may be a range because of ties.  for ease of parsing output by machine, the rank is given as a range even if the range is just one number.

(6 6 #1-1); (2 5 #2-3); (12 5 #2-3); (4 4 #4-6); (8 4 #4-6); (10 4 #4-6); (14 3 #7-9); (16 3 #7-9); (18 3 #7-9); (20 2 #10-12); (24 2 #10-12); (26 2 #10-12); (22 1 #13-14); (28 1 #13-14);

below are the top 100 differences among all pairs of the first 100 odd primes 3 through 547.

(30 60 #1-1); (60 57 #2-2); (42 54 #3-4); (90 54 #3-4); (120 49 #5-5); (6 48 #6-11); (12 48 #6-11); (24 48 #6-11); (36 48 #6-11); (66 48 #6-11); (84 48 #6-11); (18 46 #12-12); (150 45 #13-13); (48 44 #14-16); (126 44 #14-16); (210 44 #14-16); (54 42 #17-18); (78 42 #17-18); (102 40 #19-19); (72 39 #20-21); (180 39 #20-21); (96 38 #22-24); (132 38 #22-24); (168 38 #22-24); (108 37 #25-27); (114 37 #25-27); (156 37 #25-27); (144 35 #28-29); (174 35 #28-29); (138 34 #30-33); (186 34 #30-33); (240 34 #30-33); (270 34 #30-33); (10 33 #34-37); (20 33 #34-37); (70 33 #34-37); (162 33 #34-37); (216 32 #38-38); (40 31 #39-40); (192 31 #39-40); (198 30 #41-45); (204 30 #41-45); (222 30 #41-45); (228 30 #41-45); (252 30 #41-45); (14 29 #46-49); (50 29 #46-49); (56 29 #46-49); (234 29 #46-49); (80 28 #50-53); (140 28 #50-53); (264 28 #50-53); (300 28 #50-53); (4 27 #54-57); (34 27 #54-57); (246 27 #54-57); (330 27 #54-57); (22 26 #58-61); (26 26 #58-61); (44 26 #58-61); (258 26 #58-61); (2 25 #62-66); (28 25 #62-66); (38 25 #62-66); (110 25 #62-66); (276 25 #62-66); (8 24 #67-75); (16 24 #67-75); (32 24 #67-75); (52 24 #67-75); (98 24 #67-75); (100 24 #67-75); (160 24 #67-75); (294 24 #67-75); (360 24 #67-75); (46 23 #76-85); (76 23 #76-85); (82 23 #76-85); (104 23 #76-85); (130 23 #76-85); (154 23 #76-85); (170 23 #76-85); (306 23 #76-85); (312 23 #76-85); (336 23 #76-85); (68 22 #86-91); (74 22 #86-91); (86 22 #86-91); (94 22 #86-91); (112 22 #86-91); (342 22 #86-91); (62 21 #92-98); (64 21 #92-98); (88 21 #92-98); (92 21 #92-98); (190 21 #92-98); (220 21 #92-98); (282 21 #92-98); (116 20 #99-108); (118 20 #99-108); (124 20 #99-108); (146 20 #99-108); (182 20 #99-108); (200 20 #99-108); (230 20 #99-108); (288 20 #99-108); (318 20 #99-108); (390 20 #99-108);

primorials occur commonly among differences.  here are the frequencies of primorial differences among the first 100 primes, extracted from above:

(2 25 #62-66); (6 48 #6-11); (30 60 #1-1); (210 44 #14-16);

I knew in advance that primorials would be common; the motivation for this project was to see what other numbers would be common.  like primorials, the answer turned out to be smooth numbers.  finding the high frequency prime differences feels a little like decoding the music of the primes: these are the strong wavelengths and their overtones.

top differences among the first 1000 odd primes 3 through 7927:

(210 521 #1-1); (420 496 #2-2); (630 486 #3-3); (330 468 #4-4); (840 467 #5-5); (30 454 #6-6); (1050 452 #7-7); (60 447 #8-8); (390 446 #9-9); (1260 445 #10-10); (510 444 #11-12); (660 444 #11-12); (120 442 #13-13); (90 439 #14-14); (150 435 #15-15); (570 433 #16-17); (780 433 #16-17); (180 432 #18-18); (240 431 #19-20); (1470 431 #19-20); (270 430 #21-22); (300 430 #21-22); (990 423 #23-23); (360 421 #24-24); (450 418 #25-26); (690 418 #25-26); (2310 415 #27-27); (480 412 #28-28); (84 410 #29-29); (870 409 #30-31); (1680 409 #30-31); (600 405 #32-32); (42 403 #33-36); (540 403 #33-36); (750 403 #33-36); (1170 403 #33-36); (126 402 #37-37); (930 400 #38-38); (546 399 #39-39); (462 397 #40-40); (1020 394 #41-42); (1890 394 #41-42); (720 390 #43-43); (1320 389 #44-44); (714 388 #45-48); (900 388 #45-48); (960 388 #45-48); (1140 388 #45-48); (336 386 #49-51); (924 386 #49-51); (1560 386 #49-51); (168 385 #52-53); (294 385 #52-53); (66 382 #54-54); (810 380 #55-55); (798 377 #56-57); (1650 377 #56-57); (252 376 #58-59); (1080 376 #58-59); (1230 375 #60-61); (2730 375 #60-61); (504 374 #62-63); (2100 374 #62-63); (1380 371 #64-64); (378 369 #65-67); (1200 369 #65-67); (1530 369 #65-67); (1110 368 #68-68); (1980 366 #69-69); (1092 364 #70-70); (132 361 #71-73); (1386 361 #71-73); (1440 361 #71-73); (672 360 #74-76); (966 360 #74-76); (1290 360 #74-76); (264 358 #77-80); (588 358 #77-80); (858 358 #77-80); (1410 358 #77-80); (204 357 #81-81); (78 356 #82-82); (156 355 #83-84); (396 355 #83-84); (198 354 #85-86); (1350 354 #85-86); (1500 353 #87-91); (1590 353 #87-91); (1710 353 #87-91); (1860 353 #87-91); (2040 353 #87-91); (756 351 #92-92); (882 350 #93-94); (1950 350 #93-94); (234 349 #95-95); (1620 348 #96-97); (2520 348 #96-97); (306 347 #98-98); (102 345 #99-100); (1740 345 #99-100);

primorials 1000:

(2 174 #890-897); (6 343 #103-106); (30 454 #6-6); (210 521 #1-1); (2310 415 #27-27);

among the first 10000 odd primes 3 through 104743:

(2310 4371 #1-1); (4620 4306 #2-2); (2730 4260 #3-3); (5460 4127 #4-4); (3570 4118 #5-5); (6930 4107 #6-6); (3990 4088 #7-7); (210 4058 #8-8); (840 4057 #9-9); (420 4032 #10-11); (9240 4032 #10-11); (630 4027 #12-12); (8190 4011 #13-13); (4830 4000 #14-14); (1890 3993 #15-15); (1470 3992 #16-16); (1260 3980 #17-17); (1050 3977 #18-18); (7140 3969 #19-19); (1680 3964 #20-20); (2100 3949 #21-22); (11550 3949 #21-22); (2940 3923 #23-23); (2520 3917 #24-24); (7980 3908 #25-25); (4290 3902 #26-26); (3150 3897 #27-27); (10920 3886 #28-28); (3780 3878 #29-29); (6090 3876 #30-30); (6510 3873 #31-31); (4200 3866 #32-32); (3360 3865 #33-33); (4410 3855 #34-34); (5040 3832 #35-35); (13860 3820 #36-36); (10710 3815 #37-37); (7770 3800 #38-38); (5250 3795 #39-39); (8610 3774 #40-40); (5670 3769 #41-41); (9660 3760 #42-42); (330 3759 #43-43); (5880 3750 #44-44); (660 3746 #45-45); (9030 3744 #46-46); (6300 3740 #47-48); (13650 3740 #47-48); (11970 3732 #49-49); (6720 3725 #50-50); (5610 3717 #51-51); (1320 3699 #52-53); (1980 3699 #52-53); (1170 3698 #54-54); (8580 3694 #55-55); (7350 3691 #56-56); (990 3689 #57-58); (9870 3689 #57-58); (390 3686 #59-59); (1650 3678 #60-60); (16170 3676 #61-61); (7560 3672 #62-62); (9450 3665 #63-63); (780 3655 #64-64); (1560 3653 #65-65); (6270 3652 #66-66); (14280 3649 #67-67); (16380 3648 #68-68); (6630 3646 #69-69); (8400 3645 #70-70); (12180 3642 #71-71); (8820 3631 #72-72); (2640 3626 #73-73); (2970 3625 #74-74); (11130 3614 #75-75); (18480 3611 #76-76); (7590 3603 #77-77); (3300 3602 #78-78); (1950 3593 #79-79); (3630 3590 #80-80); (3960 3588 #81-81); (10080 3587 #82-82); (14490 3582 #83-83); (570 3569 #84-85); (12390 3569 #84-85); (510 3567 #86-87); (13020 3567 #86-87); (1020 3564 #88-88); (10500 3563 #89-89); (10290 3555 #90-90); (7410 3550 #91-91); (1530 3547 #92-93); (2340 3547 #92-93); (19110 3545 #94-94); (12870 3538 #95-95); (3120 3537 #96-96); (11340 3535 #97-97); (5280 3534 #98-98); (11220 3531 #99-99); (1140 3530 #100-100);

primorials 10000:

(2 1270 #12834-12857); (6 2538 #1642-1643); (30 3449 #127-127); (210 4058 #8-8); (2310 4371 #1-1); (30030 3388 #167-167);

among the first 100000 odd primes 3 through 1299721 (there are no tied ranks):

(30030 38434 #1-1); (60060 37316 #2-2); (39270 37286 #3-3); (43890 36759 #4-4); (90090 36420 #5-5); (2310 36322 #6-6); (46410 36284 #7-7); (4620 36262 #8-8); (53130 36158 #9-9); (6930 36066 #10-10); (51870 36020 #11-11); (9240 35993 #12-12); (11550 35969 #13-13); (78540 35896 #14-14); (13860 35712 #15-15); (18480 35670 #16-16); (16170 35633 #17-17); (20790 35610 #18-18); (2730 35540 #19-19); (23100 35511 #20-20); (5460 35496 #21-21); (87780 35451 #22-22); (25410 35331 #23-23); (27720 35303 #24-24); (8190 35295 #25-25); (120120 35267 #26-26); (10920 35210 #27-27); (66990 35191 #28-28); (71610 35161 #29-29); (32340 35117 #30-30); (13650 35089 #31-31); (16380 35088 #32-32); (36960 35087 #33-33); (19110 35079 #34-34); (62790 35075 #35-35); (34650 35049 #36-36); (41580 34959 #37-37); (3570 34795 #38-38); (92820 34772 #39-39); (21840 34762 #40-40); (117810 34719 #41-41); (46200 34712 #42-42); (32760 34685 #43-43); (7140 34668 #44-44); (24570 34663 #45-45); (48510 34661 #46-46); (27300 34622 #47-47); (50820 34585 #48-48); (10710 34583 #49-49); (67830 34527 #50-50); (150150 34489 #51-51); (3990 34457 #52-52); (55440 34419 #53-53); (7980 34391 #54-54); (38220 34348 #55-55); (17850 34342 #56-56); (35490 34340 #57-57); (79170 34320 #58-58); (57750 34315 #59-59); (85470 34300 #60-60); (14280 34292 #61-61); (106260 34273 #62-62); (40950 34187 #63-63); (43680 34175 #64-64); (62370 34134 #65-65); (103740 34117 #66-66); (11970 34112 #67-67); (21420 34086 #68-68); (64680 34063 #69-69); (84630 34039 #70-70); (4830 34028 #71-71); (19950 34018 #72-72); (15960 34008 #73-73); (69300 33972 #74-74); (94710 33948 #75-75); (99330 33928 #76-76); (24990 33921 #77-77); (9660 33906 #78-78); (131670 33903 #79-79); (49140 33898 #80-80); (73920 33833 #81-81); (28560 33827 #82-82); (23940 33808 #83-83); (54600 33771 #84-84); (32130 33719 #85-85); (82110 33715 #86-86); (14490 33706 #87-87); (27930 33688 #88-88); (76230 33655 #89-89); (6510 33641 #90-90); (31920 33566 #91-91); (80850 33564 #92-92); (83160 33557 #93-93); (6090 33555 #94-94); (65520 33532 #95-95); (35700 33523 #96-96); (57330 33518 #97-97); (19320 33511 #98-98); (12180 33503 #99-99); (24150 33491 #100-100);

primorials 100000:

(2 10250 #158826-158865); (6 20472 #20178-20185); (30 27434 #2003-2003); (210 32719 #161-161); (2310 36322 #6-6); (30030 38434 #1-1); (510510 25233 #5165-5165);

among the first 1000000 (10^6) odd primes 3 through 15485867 (there are no tied ranks):

(510510 340191 #1-1); (570570 335991 #2-2); (30030 331644 #3-3); (60060 330432 #4-4); (120120 329009 #5-5); (690690 328971 #6-6); (90090 328938 #7-7); (150150 327747 #8-8); (180180 327518 #9-9); (1021020 326622 #10-10); (210210 326022 #11-11); (240240 325472 #12-12); (270270 324716 #13-13); (746130 324257 #14-14); (300300 323675 #15-15); (39270 323557 #16-16); (330330 322956 #17-17); (78540 322560 #18-18); (360360 322340 #19-19); (390390 321814 #20-20); (1141140 321719 #21-21); (117810 321414 #22-22); (43890 321349 #23-23); (870870 321329 #24-24); (420420 321016 #25-25); (157080 320339 #26-26); (450450 319941 #27-27); (87780 319841 #28-28); (480480 319488 #29-29); (196350 319001 #30-30); (131670 318880 #31-31); (930930 318868 #32-32); (235620 318821 #33-33); (46410 318065 #34-34); (540540 317717 #35-35); (274890 317417 #36-36); (175560 317340 #37-37); (53130 317215 #38-38); (903210 316926 #39-39); (600600 316875 #40-40); (219450 316635 #41-41); (92820 316210 #42-42); (106260 315973 #43-43); (314160 315965 #44-44); (353430 315806 #45-45); (630630 315502 #46-46); (263340 315340 #47-47); (660660 315117 #48-48); (139230 314979 #49-49); (392700 314930 #50-50); (51870 314922 #51-51); (881790 314887 #52-52); (159390 314726 #53-53); (1531530 314692 #54-54); (185640 314104 #55-55); (307230 313971 #56-56); (66990 313959 #57-57); (103740 313791 #58-58); (1111110 313519 #59-59); (720720 313463 #60-60); (431970 313352 #61-61); (212520 313175 #62-62); (71610 313060 #63-63); (232050 312817 #64-64); (351120 312815 #65-65); (155610 312760 #66-66); (1381380 312727 #67-67); (750750 312487 #68-68); (471240 312399 #69-69); (780780 312344 #70-70); (133980 312153 #71-71); (395010 312101 #72-72); (1009470 312069 #73-73); (62790 311718 #74-74); (810810 311601 #75-75); (265650 311572 #76-76); (143220 311541 #77-77); (278460 311218 #78-78); (549780 311013 #79-79); (207480 310874 #80-80); (85470 310859 #81-81); (840840 310762 #82-82); (438900 310691 #83-83); (318780 310577 #84-84); (324870 310469 #85-85); (200970 310404 #86-86); (482790 310057 #87-87); (589050 309915 #88-88); (94710 309757 #89-89); (125580 309724 #90-90); (259350 309485 #91-91); (371280 309413 #92-92); (1231230 309364 #93-93); (214830 309355 #94-94); (371910 309351 #95-95); (900900 309325 #96-96); (99330 309314 #97-97); (628320 309159 #98-98); (170940 308915 #99-99); (526680 308795 #100-100);

primorials 1000000:

(2 86027 #1894840-1894883); (6 170910 #250608-250620); (30 228548 #26112-26114); (210 274349 #2118-2118); (2310 305046 #144-144); (30030 331644 #3-3); (510510 340191 #1-1); (9699690 141233 #642368-642376);

among the first 10^6 primes after 10^10, namely from 10^10+19 through 10^10+23010139 (there are no tied ranks):

(510510 232043 #1-1); (570570 230166 #2-2); (1021020 226921 #3-3); (690690 225737 #4-4); (1141140 224221 #5-5); (746130 222868 #6-6); (60060 221915 #7-7); (870870 221831 #8-8); (30030 221797 #9-9); (150150 221644 #10-10); (1531530 221466 #11-11); (90090 221355 #12-12); (210210 221315 #13-13); (120120 221255 #14-14); (930930 220820 #15-15); (180180 220629 #16-16); (270270 220193 #17-17); (300300 219734 #18-18); (240240 219460 #19-19); (1381380 219454 #20-20); (330330 219211 #21-21); (360360 218845 #22-22); (903210 218765 #23-23); (390390 218595 #24-24); (420420 218552 #25-25); (881790 218101 #26-26); (480480 218059 #27-27); (1111110 217837 #28-28); (450450 217779 #29-29); (1711710 217739 #30-30); (39270 217635 #31-31); (540540 217338 #32-32); (2042040 217082 #33-33); (600600 216622 #34-34); (630630 216617 #35-35); (78540 216501 #36-36); (117810 216404 #37-37); (1009470 216351 #38-38); (1231230 216241 #39-39); (720720 216045 #40-40); (196350 215994 #41-41); (157080 215829 #42-42); (43890 215797 #43-43); (660660 215533 #44-44); (235620 215460 #45-45); (1492260 215392 #46-46); (1291290 215335 #47-47); (750750 215144 #48-48); (780780 214788 #49-49); (274890 214773 #50-50); (810810 214767 #51-51); (131670 214740 #52-52); (87780 214737 #53-53); (314160 214733 #54-54); (840840 214580 #55-55); (353430 214368 #56-56); (1138830 214271 #57-57); (900900 214269 #58-58); (175560 213720 #59-59); (219450 213694 #60-60); (1741740 213676 #61-61); (46410 213530 #62-62); (392700 213441 #63-63); (307230 213434 #64-64); (1217370 213430 #65-65); (1067430 213417 #66-66); (431970 213342 #67-67); (263340 213308 #68-68); (960960 213186 #69-69); (92820 213075 #70-70); (471240 213069 #71-71); (1411410 213067 #72-72); (351120 212893 #73-73); (53130 212760 #74-74); (589050 212729 #75-75); (990990 212645 #76-76); (2072070 212598 #77-77); (159390 212486 #78-78); (139230 212460 #79-79); (106260 212415 #80-80); (1051050 212366 #81-81); (395010 212241 #82-82); (549780 212217 #83-83); (1272810 212202 #84-84); (2282280 212191 #85-85); (1081080 212140 #86-86); (185640 212057 #87-87); (1861860 211670 #88-88); (51870 211619 #89-89); (1591590 211566 #90-90); (628320 211537 #91-91); (265650 211535 #92-92); (212520 211497 #93-93); (232050 211469 #94-94); (324870 211438 #95-95); (1171170 211347 #96-96); (438900 211271 #97-97); (482790 211263 #98-98); (278460 211262 #99-99); (1201200 211197 #100-100);

primorials at 10^10:

(2 57382 #2903318-2903469); (6 114833 #392781-392814); (30 152755 #45239-45242); (210 183875 #3655-3655); (2310 204068 #307-307); (30030 221797 #9-9); (510510 232043 #1-1); (9699690 145066 #85780-85787);

among the first 10^6 primes after 10^100, namely 10^100+267 through 10^100+230441007:

(9699690 24034 #1-1); (11741730 23765 #2-2); (1531530 23622 #3-3); (1021020 23590 #4-4); (2282280 23564 #5-5); (3063060 23542 #6-6); (2552550 23519 #7-7); (510510 23497 #8-8); (2042040 23482 #9-9); (1141140 23475 #10-10); (570570 23462 #11-11); (1711710 23418 #12-12); (3423420 23406 #13-13); (690690 23388 #14-14); (3573570 23324 #15-15); (4594590 23270 #16-16); (4564560 23239 #17-18); (13123110 23239 #17-18); (4084080 23237 #19-19); (1381380 23115 #20-21); (5705700 23115 #20-21); (3993990 23109 #22-22); (6276270 23090 #23-23); (2852850 23083 #24-24); (14804790 23037 #25-25); (4144140 23006 #26-26); (870870 22993 #27-27); (6636630 22968 #28-28); (7147140 22960 #29-29); (6126120 22948 #30-30); (8678670 22946 #31-31); (5135130 22937 #32-32); (7417410 22929 #33-33); (2072070 22922 #34-34); (5615610 22916 #35-35); (19399380 22908 #36-36); (15825810 22900 #37-37); (6846840 22887 #38-38); (2984520 22881 #39-39); (746130 22880 #40-40); (7987980 22858 #41-41); (7657650 22852 #42-42); (3730650 22843 #43-43); (2238390 22838 #44-44); (6216210 22837 #45-45); (3453450 22835 #46-46); (1741740 22821 #47-47); (9189180 22820 #48-48); (4834830 22799 #49-49); (1111110 22790 #50-50); (930930 22756 #51-51); (2762760 22747 #52-52); (5105100 22735 #53-54); (8168160 22735 #53-54); (5222910 22732 #55-55); (3723720 22713 #56-56); (1411410 22709 #57-57); (5969040 22708 #58-58); (2792790 22703 #59-59); (10840830 22702 #60-60); (2612610 22696 #61-61); (1231230 22693 #62-62); (6906900 22674 #63-63); (3333330 22672 #64-64); (903210 22667 #65-65); (1492260 22640 #66-66); (1591590 22625 #67-67); (8288280 22624 #68-68); (2222220 22623 #69-69); (4444440 22619 #70-70); (16546530 22607 #71-71); (9129120 22587 #72-72); (4354350 22586 #73-74); (5585580 22586 #73-74); (3483480 22577 #75-75); (1861860 22575 #76-76); (10720710 22566 #77-77); (5225220 22565 #78-78); (2645370 22551 #79-79); (7597590 22542 #80-80); (8558550 22538 #81-81); (5525520 22537 #82-82); (12252240 22534 #83-83); (10210200 22526 #84-84); (60060 22512 #85-86); (11231220 22512 #85-86); (881790 22510 #87-87); (12762750 22500 #88-88); (1009470 22485 #89-89); (2132130 22482 #90-90); (13783770 22472 #91-91); (1291290 22468 #92-92); (1806420 22457 #93-94); (2709630 22457 #93-94); (2277660 22456 #95-95); (2582580 22448 #96-96); (2822820 22447 #97-97); (4654650 22446 #98-98); (2492490 22435 #99-99); (11981970 22425 #100-100);

primorials at 10^100:

(2 5767 #28706071-28717948); (6 11484 #3922267-3924636); (30 15466 #384556-384851); (210 18481 #32555-32582); (2310 20373 #3393-3398); (30030 22276 #135-135); (510510 23497 #8-8); (9699690 24034 #1-1); (223092870 815 #102930136-102945303);

future work: 10^1000.

[egfgwujd] twin primes adjacent to highly composite numbers

below are some numbers N with many divisors and for which N-1 and N+1 are both prime, i.e., twin primes.

the suffix A indicates a highly composite number (sets a new record for number of divisors among all numbers, not necessarily those between twin primes); B similarly indicates it ties the record for number of divisors (largely composite number); C indicates a new second place in number of divisors; D ties second place number of divisors.

4/A 6/A 12/A 18/A 30/B 42/D 60/A 72/B 108/B 180/A 240/A 420/B 432/D 600/A 660/B 1320/D 2340/D 3360/A 5280/D 5880/D 6300/C 7560/A 9240/A 21840/D 35280/D 42840/B 55440/A 65520/A 92400/D 100800/C 110880/A 128520/B 180180/D 415800/B 453600/D 514080/D 526680/D 540540/D 1867320/D 1912680/D 1921920/D 1940400/C 2489760/D 6652800/D 6846840/D 7068600/D 28828800/B 31600800/B 34594560/D 85765680/D 100900800/C 121080960/D 232792560/A 287567280/D 397837440/D 634888800/D 845404560/D 1259818560/D 1470268800/B 1574773200/D 6299092800/D 10708457760/D 12681068400/B 23827003200/D 32125373280/B 32590958400/C 33816182400/A 34918884000/B 40156716600/B 73329656400/A 135019684800/D 216497080800/B 439977938400/D 449755225920/B 578256719040/D 2126560035600/D 2835413380800/B 2971597028400/D 6278415343200/B 18632716502400/A 18835246029600/C 19275223968000/B 59753194300800/B 62403537596400/D 69712060017600/B 92199821313600/B 130429015516800/A 279490747536000/C 313704270079200/B 448148957256000/D 838472242608000/D 1382997319704000/D 1402111916805600/B 1498809290378400/D

investigated 1169 numbers from 1 to 1516237305382800 in those categories (A, B, C, D) previously generated.  95 numbers (above) are surrounded by twin primes.

inspired by some numbers between twin primes being quite smooth.  if searching for large twin primes (not sure why one would want to do this), is it profitable to restrict to those bracketing a smooth number?  (seems yes.)  if so, how smooth?  perhaps Pierpont, or some other sequence of very smooth numbers.

[bvqqdwkp] Pierpont twin primes

below are the 73 twin primes of the form (2^a * 3^b plus and minus 1) less than 2^10000 ~= 10^3010.

(2^2 * 3^0 +- 1), (2^1 * 3^1 +- 1), (2^2 * 3^1 +- 1), (2^1 * 3^2 +- 1), (2^3 * 3^2 +- 1), (2^2 * 3^3 +- 1), (2^6 * 3^1 +- 1), (2^4 * 3^3 +- 1), (2^7 * 3^2 +- 1), (2^5 * 3^4 +- 1), (2^6 * 3^7 +- 1), (2^3 * 3^10 +- 1), (2^18 * 3^1 +- 1), (2^12 * 3^5 +- 1), (2^2 * 3^15 +- 1), (2^18 * 3^5 +- 1), (2^21 * 3^4 +- 1), (2^24 * 3^5 +- 1), (2^27 * 3^4 +- 1), (2^30 * 3^7 +- 1), (2^33 * 3^8 +- 1), (2^43 * 3^2 +- 1), (2^32 * 3^9 +- 1), (2^36 * 3^7 +- 1), (2^11 * 3^24 +- 1), (2^31 * 3^12 +- 1), (2^43 * 3^8 +- 1), (2^32 * 3^15 +- 1), (2^50 * 3^9 +- 1), (2^63 * 3^2 +- 1), (2^66 * 3^25 +- 1), (2^79 * 3^20 +- 1), (2^99 * 3^10 +- 1), (2^57 * 3^64 +- 1), (2^82 * 3^63 +- 1), (2^148 * 3^27 +- 1), (2^63 * 3^88 +- 1), (2^56 * 3^99 +- 1), (2^211 * 3^2 +- 1), (2^275 * 3^16 +- 1), (2^287 * 3^10 +- 1), (2^90 * 3^169 +- 1), (2^148 * 3^135 +- 1), (2^298 * 3^51 +- 1), (2^160 * 3^141 +- 1), (2^363 * 3^52 +- 1), (2^134 * 3^231 +- 1), (2^49 * 3^320 +- 1), (2^529 * 3^44 +- 1), (2^264 * 3^419 +- 1), (2^960 * 3^143 +- 1), (2^541 * 3^476 +- 1), (2^988 * 3^207 +- 1), (2^1015 * 3^332 +- 1), (2^1440 * 3^97 +- 1), (2^1295 * 3^324 +- 1), (2^979 * 3^738 +- 1), (2^258 * 3^1493 +- 1), (2^637 * 3^1320 +- 1), (2^2320 * 3^333 +- 1), (2^1036 * 3^1167 +- 1), (2^2815 * 3^188 +- 1), (2^1063 * 3^1440 +- 1), (2^180 * 3^2251 +- 1), (2^888 * 3^2033 +- 1), (2^300 * 3^2819 +- 1), (2^2176 * 3^2175 +- 1), (2^4014 * 3^1879 +- 1), (2^280 * 3^4311 +- 1), (2^6228 * 3^571 +- 1), (2^6981 * 3^560 +- 1), (2^3505 * 3^2892 +- 1), (2^5899 * 3^2570 +- 1)

is this a statistically high yield of twin primes?

also previously.

[aoypoxfn] factorization challenges easy to type

each composite number below consists of repetitions of the digit string 1234567890 except the final few digits.  the smaller factors are large enough that quadratic sieve is probably more efficient than elliptic curve method (ECM).

previously similar for base 2, assuming you have an expression parser.

40 digits: 1234567890123456789012345678901234567759 = 5284744045008013 * 233609779321220631386443

50 digits: 12345678901234567890123456789012345678901234567633 = 92279162895906968524229 * 133786203881810018387538077

60 digits: 123456789012345678901234567890123456789012345678901234567831 = 87988560978576315942371700757 * 1403100444413510291678340665083

70 digits: 1234567890123456789012345678901234567890123456789012345678901234566713 = 61208527479308458227554760703 * 20169867516267279020392646921656369545671

80 digits: 12345678901234567890123456789012345678901234567890123456789012345678901234567751 = 575099128882847693294536030177708942207 * 21467045038333698701383939676398791777593

90 digits: 123456789012345678901234567890123456789012345678901234567890123456789012345678901234567597 = 40166540925345636983244707230202848993 * 3073622626399545288925403158158369054873454157207629

[eoaqgkaq] proportion of RSA numbers

define an RSA number to be a number that factors into exactly two prime numbers, the larger being less than twice the smaller.

isrsa(n)=my(f=factorint(n)); my(s=matsize(f)); if(2!=s[1], return(0), if(f[1,2]!=1 || f[2,2]!=1, return(0), if(f[1,1]*2<f[2,1], return(0)))); 1;

here are the first few RSA numbers:

? for(i=0, 256, if(isrsa(i), print(i," ",factorint(i))))
6 [2, 1; 3, 1]
15 [3, 1; 5, 1]
35 [5, 1; 7, 1]
77 [7, 1; 11, 1]
91 [7, 1; 13, 1]
143 [11, 1; 13, 1]
187 [11, 1; 17, 1]
209 [11, 1; 19, 1]
221 [13, 1; 17, 1]
247 [13, 1; 19, 1]

for a given range of numbers, what proportion are RSA numbers?  this seems like this could be estimated analytically (perhaps with the Dickman function, for related work, see Tao), but we do Monte Carlo in Pari/GP, using parfor for parallel computation.

in order to speed up computation by a factor of 2, we assume all even numbers are not RSA numbers.  however, when testing small numbers, 6 needs to be special cased.  even though printing is done in the sequential block of parfor, the results do not come out in order.

the first few results below can be verified with the above list of RSA numbers.  among the 2-bit numbers with most significant bit set, namely 2 and 3, none are RSA numbers.  among the 5-bit numbers with MSB set, 16-31, none are RSA numbers.  among the 32 6-bit numbers with MSB set, 32-63, 1 (35) is an RSA number: 1e6/32 = 31250.  among the 8 4-bit numbers with MSB set, 8-15, 1 (15) is an RSA number: 1e6/8 = 125000.  among the 128 8-bit numbers with MSB set, 128-255, 5 (143, 187, 209, 221, 247) are RSA numbers: 1e6/128*5 = 39062.5.  among the 1-bit number with MSB set, 1, none are RSA numbers.  among the 64 7-bit numbers with MSB set, 64-127, 2 (35, 77) are RSA numbers: 1e6/64*2 = 31250.  among the 4 3-bit numbers with MSB set, 4-7, 1 (6) is an RSA number: 1e6/4 = 250000.

first batch (actually done second), up to 55 bits:

? gettime; export(isrsa); my(ncount=1000000); parfor(bitsize=1, 55, my(s=0); for(i=1, ncount, my(n=random(2^(bitsize-1))+2^(bitsize-1)); if((n!=6)&&(n%2==0), next); if(isrsa(n), s+=1)); s, x, print("dataline "bitsize" "x" "ncount" "gettime))
dataline 2 0 1000000 18573
dataline 5 0 1000000 1904
dataline 6 31267 1000000 721
dataline 4 124853 1000000 161
dataline 8 39461 1000000 429
dataline 1 0 1000000 657
dataline 7 31457 1000000 619
dataline 3 249848 1000000 3514
dataline 10 19328 1000000 15600
dataline 9 19521 1000000 358
dataline 13 15826 1000000 771
dataline 11 18540 1000000 435
dataline 12 17816 1000000 656
dataline 15 11376 1000000 1572
dataline 14 12527 1000000 70
dataline 16 10891 1000000 2589
dataline 17 9477 1000000 15102
dataline 20 7257 1000000 2855
dataline 18 8500 1000000 1187
dataline 19 7705 1000000 1742
dataline 21 6747 1000000 2350
dataline 22 5958 1000000 1498
dataline 23 5487 1000000 2263
dataline 24 5089 1000000 2358
dataline 25 4625 1000000 15038
dataline 26 4297 1000000 5640
dataline 27 3963 1000000 2492
dataline 28 3790 1000000 3167
dataline 29 3613 1000000 3506
dataline 30 3317 1000000 6984
dataline 31 3043 1000000 6711
dataline 32 2769 1000000 8230
dataline 33 2616 1000000 23971
dataline 34 2583 1000000 18296
dataline 35 2467 1000000 10707
dataline 36 2216 1000000 8008
dataline 37 2173 1000000 17621
dataline 38 2078 1000000 16277
dataline 39 1908 1000000 25297
dataline 40 1863 1000000 22039
dataline 41 1790 1000000 39006
dataline 42 1694 1000000 42367
dataline 43 1593 1000000 38054
dataline 44 1524 1000000 35728
dataline 45 1499 1000000 49830
dataline 46 1410 1000000 48127
dataline 47 1351 1000000 62108
dataline 48 1282 1000000 65871
dataline 49 1188 1000000 76204
dataline 50 1231 1000000 71772
dataline 51 1122 1000000 66634
dataline 52 1074 1000000 44500
dataline 53 1029 1000000 56539
dataline 54 1007 1000000 37354
dataline 55 954 1000000 22767
cpu time = 17min, 8,807 ms, real time = 5min, 50,115 ms.

for example, the last line above should be read: we randomly sampled 1000000 55-bit numbers with MSB set.  954 were RSA numbers = 0.000954 = 0.0954% .  22767 milliseconds of CPU time had been expended since the previous line was printed (though gettime is mostly meaningless in this parallel computation).

second batch:

? gettime; export(isrsa); my(ncount=1000000); parfor(bitsize=56, +oo, my(s=0); for(i=1, ncount, my(n=random(2^(bitsize-1))+2^(bitsize-1)); if(n%2==0, next); if(isrsa(n), s+=1)); s, x, print("dataline "bitsize" "x" "ncount" "gettime))
dataline 56 928 1000000 967649
dataline 57 913 1000000 139615
dataline 58 887 1000000 154693
dataline 60 841 1000000 74465
dataline 59 837 1000000 107296
dataline 61 850 1000000 11812
dataline 62 763 1000000 70641
dataline 63 726 1000000 78389
dataline 64 722 1000000 1068514
dataline 65 690 1000000 446623
dataline 66 638 1000000 266677
dataline 67 694 1000000 211371
dataline 68 641 1000000 253129
dataline 69 619 1000000 164697
dataline 70 557 1000000 221582
dataline 71 577 1000000 253060
dataline 72 583 1000000 1208599
dataline 73 511 1000000 584578
dataline 74 502 1000000 448724
dataline 75 498 1000000 380748
dataline 76 499 1000000 391445
dataline 77 470 1000000 382709
dataline 78 471 1000000 403028
dataline 79 483 1000000 458870
dataline 80 495 1000000 1413907
dataline 81 431 1000000 791296
dataline 82 436 1000000 694658
dataline 83 411 1000000 649589
dataline 84 412 1000000 550870
dataline 85 399 1000000 621967
dataline 86 407 1000000 650454
dataline 87 399 1000000 717952
dataline 88 392 1000000 1735925
dataline 89 342 1000000 1041960
dataline 90 352 1000000 996877
dataline 91 346 1000000 1002713
dataline 92 341 1000000 860047
dataline 93 304 1000000 955470
dataline 94 333 1000000 1012760
dataline 95 329 1000000 1054538
dataline 96 299 1000000 2080883
dataline 97 299 1000000 1574054
dataline 98 283 1000000 1497025
dataline 99 300 1000000 1541806
dataline 100 313 1000000 1266607
dataline 101 287 1000000 1577068
dataline 102 289 1000000 1557525
dataline 103 265 1000000 1650203
dataline 104 299 1000000 2721996
dataline 105 268 1000000 2171018
dataline 106 261 1000000 2138428
dataline 107 248 1000000 2464619
dataline 108 273 1000000 2069371
dataline 109 233 1000000 2549832
dataline 110 228 1000000 2507602
dataline 111 242 1000000 2683309
dataline 112 225 1000000 3787277
dataline 113 223 1000000 3385594
dataline 114 181 1000000 3457664
dataline 115 212 1000000 3773621
dataline 116 235 1000000 3241294
dataline 117 212 1000000 4039748
dataline 118 196 1000000 4076002
dataline 119 197 1000000 4216571
dataline 120 191 1000000 6046672
dataline 121 205 1000000 5242164
dataline 122 193 1000000 5826027
dataline 123 184 1000000 5985150
dataline 124 193 1000000 5220699
dataline 125 183 1000000 6394398
dataline 126 166 1000000 6804860
dataline 127 182 1000000 7052941
dataline 128 199 1000000 9081431
dataline 129 164 1000000 9493346
dataline 130 161 1000000 9527733
dataline 131 151 1000000 10311288
dataline 132 177 1000000 9186555
dataline 133 169 1000000 11124547
dataline 134 178 1000000 12562868
dataline 135 154 1000000 12741479
dataline 136 167 1000000 15479663
dataline 137 157 1000000 15436601
dataline 138 155 1000000 16690681
dataline 139 148 1000000 17277977
dataline 140 137 1000000 19504895
dataline 141 162 1000000 18781695
dataline 142 138 1000000 22060326
dataline 143 152 1000000 22222123
dataline 144 134 1000000 26356486
dataline 145 150 1000000 24985983
dataline 146 159 1000000 27665757
dataline 147 155 1000000 29276993
dataline 148 124 1000000 31849872
dataline 149 127 1000000 30683069
dataline 150 126 1000000 35491505
*** parfor: user interrupt after 163h, 51min, 56,082 ms cpu time, 21h, 51min, 54,044 ms real time

[novhsvil] Pennsylvania municipalities and CDPs

Pennsylvania has a very large number of municipalities (2560), with many duplicated and similar names.  we work with this spreadsheet https://dced.pa.gov/wp-content/themes/business2015/csv/municipalities.csv linked from https://dced.pa.gov/local-government/municipal-statistics/municipalities/.

it is probably not authoritative, with inconsistencies such as both "St Clair" and "St. Clair".  it still has "McCandless Township", even though the township officially changed its name (but not type) to "The Town of McCandless".  (it's not clear whether it has the authority to change its name, or that the state has to recognize the name change.)

weird municipality names: "S.N.P.J." stands for "Slovenska Narodna Podporna Jednota".  Valley-Hi is the only hyphenated municipality in the spreadsheet.  "Wilkes Barre" is not hyphenated in the spreadsheet.

two spaces in Harris..Township and Metal..Township.

the former name of Waverly is given in parentheses: "(Abington Township)".  this is the only instance of parentheses.

there is both "Mt Joy" and "Mount Joy".

listed below as "County:Name" are occurrences of the same name more than once within a county.  in each case, the duplication is with a township and one of {city, borough}.  there are no triplications (or more).  there are no instances of duplications of "Name:Type" within a county.

"Allegheny:Baldwin", "Allegheny:Elizabeth", "Allegheny:Springdale", "Armstrong:Kittanning", "Beaver:Darlington", "Beaver:Rochester", "Bedford:Bedford", "Bedford:Hopewell", "Bedford:Woodbury", "Blair:Tyrone", "Bradford:Athens", "Bradford:Burlington", "Bradford:Canton", "Bradford:Monroe", "Bradford:Rome", "Bradford:Towanda", "Bradford:Troy", "Bradford:Wyalusing", "Bucks:Bristol", "Bucks:Doylestown", "Bucks:New Britain", "Bucks:Newtown", "Butler:Butler", "Butler:Connoquenessing", "Butler:Fairview", "Butler:Slippery Rock", "Cambria:Cresson", "Cambria:Gallitzin", "Cambria:Portage", "Cambria:Summerhill", "Centre:Howard", "Centre:Snow Shoe", "Chester:Honey Brook", "Clarion:Clarion", "Clarion:Knox", "Clearfield:Burnside", "Clinton:Beech Creek", "Columbia:Benton", "Columbia:Briar Creek", "Columbia:Catawissa", "Crawford:Venango", "Crawford:Woodcock", "Cumberland:Shippensburg", "Dauphin:Halifax", "Dauphin:Lykens", "Delaware:Chester", "Delaware:Darby", "Elk:Ridgway", "Erie:Girard", "Erie:Mckean", "Erie:North East", "Erie:Waterford", "Fayette:Brownsville", "Fayette:Connellsville", "Fayette:Dunbar", "Forest:Tionesta", "Greene:Jefferson", "Lackawanna:Carbondale", "Lancaster:Ephrata", "Lancaster:Lancaster", "Lancaster:Manheim", "Lancaster:Strasburg", "Luzerne:Conyngham", "Luzerne:Dallas", "Luzerne:Exeter", "Luzerne:Kingston", "Luzerne:Nescopeck", "Luzerne:Pittston", "Luzerne:Plymouth", "Luzerne:Wilkes Barre", "Lycoming:Muncy", "Mckean:Bradford", "Mckean:Eldred", "Mercer:Sandy Lake", "Montgomery:Hatfield", "Northampton:Bethlehem", "Northumberland:Mt Carmel", "Northumberland:Shamokin", "Perry:Liverpool", "Pike:Milford", "Potter:Oswayo", "Potter:Ulysses", "Schuylkill:Pine Grove", "Schuylkill:Tremont", "Somerset:Addison", "Somerset:Paint", "Somerset:Somerset", "Sullivan:Laporte", "Susquehanna:Great Bend", "Susquehanna:New Milford", "Susquehanna:Oakland", "Susquehanna:Thompson", "Tioga:Liberty", "Tioga:Tioga", "Tioga:Westfield", "Warren:Sugar Grove", "Westmoreland:Derry", "Westmoreland:Donegal", "Westmoreland:Ligonier", "Westmoreland:Mt Pleasant", "Westmoreland:Penn", "Wyoming:Meshoppen", "Wyoming:Nicholson", "Wyoming:Tunkhannock", "York:Dover", "York:Manchester", "York:Shrewsbury", "York:Windsor", "York:York"

name duplications within a county of the same name, so triplication (subset of above): Bedford, Butler, Clarion, Lancaster, Somerset, Tioga, York.

the following municipalities have names with a space in them, but there is also another municipality omitting the space: "Black Lick", "Cherry Tree", "French Creek", "Mill Creek" "Oil Creek", "Pine Grove", "Sandy Creek"

the following municipalities have names ending in "City" but are not cities: "Broad Top City", "Central City", "Dickson City", "Ellwood City", "Evans City", "Fayette City", "Ford City", "Forest City", "Grove City", "Homer City", "Karns City", "Lake City", "Mahanoy City", "Spring City", "Tower City", "Union City".  they are all (currently) boroughs.  (someday, they might upgrade to cities.  someday, they might drop "City" from their name.)

"Oil City" is (I think) the only city whose name ends in City; however, the spreadsheet includes entries such as "Philadelphia City" and "Pittsburgh City" suggesting the last word in city names should be truncated, in which case the entry "Oil City" (not "Oil City City") would be truncated to just "Oil".

FOO (typically as FOO Township) and FOO City (as a borough) both exist: "Broad Top", "Fayette", "Grove", "Homer", "Lake", "Mahanoy", "Spring", "Union"

"East Pennsboro" and "West Pennsboro" are the only two municipalities whose names end in "-boro" but are not boroughs.  they are townships.

"Marlborough", "East Malborough", and "West Marlborough" are the only municipalities which end in "-borough".  they are townships.

FOO and FOOtown both exist: "Adams", "Allen", "Beaver", "Carroll", "Dallas", "Dubois", "Elizabeth", "Franklin", "Jones", "Lewis", "Mifflin", "Richland", "Snyder", "Thompson", "Union", "Watson", "West", "Williams"

FOO and FOOstown both exist: "Abbott", "Brown", "Cooper", "Jenner", "Miller", "Stewart", "Wright", "Young"

FOO and FOOton both exist: "Elder", "Falls", "Forks", "Hazle", "Lehigh", "Logan", "Palmer", "Smith", "Wallace"

some other near duplicates in this family: Charleston-Charlestown, Easton-Easttown, Newton-Newtown, Middleton-Middletown

FOO and FOOville both exist: "Bern", "Bigler", "Cass", "Center", "Clinton", "College", "Conneaut", "Elizabeth", "Forks", "Girard", "Green", "Knox", "Lawrence", "Logan", "Manor", "Mead", "Monroe", "Morris", "Ohio", "Orange", "Pleasant", "Plum", "Rose", "Shippen", "Spring", "Turbot", "Union", "Washington", "Wells", "Worth"

FOO and FOOsville both exist: "Blair", "Brown", "Clark", "Clay", "Duncan", "Miller", "Montour", "Porter", "St Clair", "Wright", "Young"

FOO and FOOburg both exist: "Adams", "Bloss", "Fox", "Lewis", "Mifflin", "Miles", "Peters", "Watts", "Wilkins", "Williams"

FOO and FOOsburg both exist: "Bloom", "Cooper", "Green", "Mercer", "Miller", "Penn", "Shippen", "Shirley", "Stroud", "Wayne"

of the many municipalities which end in "-burg" or "-burgh", most are boroughs, but the following are not.  Shippensburg and Strasburg are townships.  Harrisburg, Pittsburgh (the only instance of "-burgh"), and Greensburg are cities.  Bloomsburg is the only "town" in Pennsylvania, but its government resembles that of a borough.

FOO and FOO Creek both exist: "Black", "Elk", "Falls", "Licking", "Muncy", "Pine", "Sandy", "Spring", "Wolf"

FOO and FOOdale both exist: "Carbon", "Coal", "Colebrook", "Scott", "Spring", "Union"

FOO and FOO Hills both exist: "Braddock", "Green", "Jefferson", "Penn", "Pleasant", "Sewickley"

FOO and FOO Heights both exist: "Ben Avon", "Chester", "Clifton", "Marion", "Patterson", "Sewickley"

the most common names are "Washington" (23), "Franklin" (18), "Jackson" (18), "Union" (18), "Penn" (14), "Jefferson" (11), "Liberty" (10), "Perry" (9), "Springfield" (9), "Wayne" (9), "Monroe" (8), "Pine" (8), "Eldred" (7), "Greene" (7), "Hopewell" (7), "Porter" (7), "Allegheny" (6), "Beaver" (6), "Fairview" (6), "Hanover" (6), "Mt Pleasant" (6), "Richland" (6).

the most common prefix words are "West" (76), "East" (65), "North" (40), "New" (39), "South" (38), "Upper" (37), "Lower" (32).  some other (subjectively) generic prefixes: "Northern", "Northeast", "Southwest", "Port", "Mt", "Mount", "Middle", "Little"

the most common suffix words are "Creek" (32), "City" (17), "Grove" (13), "Lake" (9), "Valley" (9), "Hill" (8), "Park" (8), "Heights" (7), "Hills" (7), "Springs" (7).  note that some of these are also common as suffixes not a distinct word: "-creek", "-grove", "-hill".  and "Brothersvalley".  some other (subjectively) generic suffix words: "Heights", "Center", "Hills", "Haven", "Village"

after elimination of duplicates (some subjective decisions made), here are a list of 1886 names:

Abbott, Abbottstown, Abington, Adams, Adamsburg, Adamstown, Addison, Akron, Alba, Albany, Albion, Alburtis, Aldan, Aleppo, Alexandria, Aliquippa, Allegany, Allegheny, Allen, Allenport, Allentown, Allison, Alsace, Altoona, Ambler, Ambridge, Amity, Amwell, Annin, Annville, Anthony, Antis, Antrim, Apolacon, Apollo, Applewold, Ararat, Archbald, Arendtsville, Armagh, Armenia, Armstrong, Arnold, Arona, Ashland, Ashley, Ashville, Aspinwall, Aston, Asylum, Atglen, Athens, Atwood, Auburn, Austin, Avalon, Avis, Avoca, Avondale, Avonmore, Ayr, Baden, Bald Eagle, Baldwin, Bally, Bangor, Banks, Barkeyville, Barnett, Barr, Barree, Barrett, Barry, Bart, Bastress, Bath, Beale, Beallsville, Bear Creek, Bear Creek Village, Bear Lake, Beaver, Beaver Falls, Beaver Meadows, Beavertown, Beccaria, Bechtelsville, Bedford, Bedminster, Beech Creek, Belfast, Bell, Bell Acres, Belle Vernon, Bellefonte, Bellevue, Bellwood, Ben Avon, Ben Avon Heights, Bendersville, Benezette, Benner, Bensalem, Benson, Bentleyville, Benton, Berlin, Bern, Bernville, Berrysburg, Berwick, Bessemer, Bethany, Bethel, Bethel Park, Bethlehem, Big Beaver, Big Run, Bigler, Biglerville, Bingham, Birdsboro, Birmingham, Black, Black Creek, Black Lick, Blacklick, Blain, Blaine, Blair, Blairsville, Blakely, Blawnox, Bloom, Bloomfield, Blooming Grove, Blooming Valley, Bloomsburg, Bloss, Blossburg, Blythe, Boggs, Bolivar, Bonneauville, Boswell, Bowmanstown, Boyertown, Brackenridge, Braddock, Braddock Hills, Bradford, Bradford Woods, Brady, Bradys Bend, Braintrim, Branch, Bratton, Brecknock, Brentwood, Briar Creek, Bridgeport, Bridgeton, Bridgeville, Bridgewater, Brighton, Brisbin, Bristol, Broad Top, Broad Top City, Brockway, Brokenstraw, Brookfield, Brookhaven, Brooklyn, Brookville, Brothersvalley, Brown, Brownstown, Brownsville, Bruin, Brush Creek, Brush Valley, Bryn Athyn, Buck, Buckingham, Buffalo, Buffington, Bullskin, Burgettstown, Burlington, Burnham, Burnside, Burrell, Bushkill, Butler, Cadogan, Caernarvon, California, Callensburg, Callery, Callimont, Caln, Cambria, Cambridge, Cambridge Springs, Camp Hill, Canaan, Canal, Canoe, Canonsburg, Canton, Carbon, Carbondale, Carlisle, Carmichaels, Carnegie, Carroll, Carroll Valley, Carrolltown, Cascade, Cass, Cassandra, Casselman, Cassville, Castanea, Castle Shannon, Catasauqua, Catawissa, Catharine, Cecil, Center, Centerport, Centerville, Central City, Centralia, Centre, Centre Hall, Ceres, Chadds Ford, Chalfant, Chalfont, Chambersburg, Chanceford, Chapman, Charleroi, Charleston, Charlestown, Chartiers, Chatham, Cheltenham, Cherry, Cherry Grove, Cherry Ridge, Cherry Tree, Cherry Valley, Cherryhill, Cherrytree, Chest, Chest Springs, Chester, Chester Heights, Chester Hill, Chestnuthill, Cheswick, Chicora, Chippewa, Choconut, Christiana, Churchill, Clairton, Clara, Clarendon, Clarion, Clark, Clarks Green, Clarks Summit, Clarksville, Clay, Claysville, Clearfield, Cleona, Cleveland, Clifford, Clifton, Clifton Heights, Clinton, Clintonville, Clover, Clymer, Coal, Coal Center, Coaldale, Coalmont, Coalport, Coatesville, Cochranton, Codorus, Cogan House, Cokeburg, Cold Spring, Colebrook, Colebrookdale, Colerain, College, Collegeville, Colley, Collier, Collingdale, Columbia, Columbus, Colwyn, Concord, Conemaugh, Conestoga, Conewago, Conewango, Confluence, Conneaut, Conneaut Lake, Conneautville, Connellsville, Connoquenessing, Conoy, Conshohocken, Conway, Conyngham, Cook, Cooke, Coolbaugh, Coolspring, Cooper, Coopersburg, Cooperstown, Coplay, Coraopolis, Cornplanter, Cornwall, Corry, Corsica, Corydon, Coudersport, Courtdale, Covington, Cowanshannock, Crafton, Cranberry, Cranesville, Crawford, Creekside, Crescent, Cresson, Cressona, Cromwell, Cross Creek, Cross Roads, Croyle, Cumberland, Cumberland Valley, Cummings, Cumru, Curtin, Curwensville, Cussewago, Daisytown, Dale, Dallas, Dallastown, Dalton, Damascus, Danville, Darby, Darlington, Daugherty, Dauphin, Davidson, Dawson, Dayton, Dean, Decatur, Deemston, Deer Creek, Deer Lake, Deerfield, Delano, Delaware, Delaware Water Gap, Delmar, Delmont, Delta, Dennison, Denver, Derry, Dickinson, Dickson City, Dillsburg, Dimock, Dingman, District, Donegal, Donora, Dormont, Dorrance, Douglass, Dover, Downingtown, Doylestown, Dravosburg, Dreher, Driftwood, Drumore, Dublin, Dubois, Duboistown, Dudley, Dunbar, Duncan, Duncannon, Duncansville, Dunkard, Dunlevy, Dunmore, Dunnstable, Dupont, Duquesne, Durham, Duryea, Dushore, Dyberry, Eagles Mere, Earl, East Allen, East Bangor, East Berlin, East Bethlehem, East Bradford, East Brady, East Brandywine, East Brunswick, East Buffalo, East Butler, East Caln, East Cameron, East Carroll, East Chillisquaque, East Cocalico, East Conemaugh, East Coventry, East Deer, East Donegal, East Drumore, East Earl, East Fairfield, East Fallowfield, East Finley, East Franklin, East Goshen, East Greenville, East Hanover, East Hempfield, East Hopewell, East Huntingdon, East Keating, East Lackawannock, East Lampeter, East Lansdowne, East Mahoning, East Manchester, East Marlborough, East Mckeesport, East Mead, East Nantmeal, East Norriton, East Norwegian, East Nottingham, East Penn, East Pennsboro, East Petersburg, East Pikeland, East Pittsburgh, East Prospect, East Providence, East Rochester, East Rockhill, East Side, East St Clair, East Stroudsburg, East Taylor, East Union, East Vandergrift, East Vincent, East Washington, East Wheatfield, East Whiteland, Easton, Easttown, Eastvale, Eaton, Eau Claire, Ebensburg, Economy, Eddystone, Eden, Edgewood, Edgeworth, Edgmont, Edinboro, Edwardsville, Ehrenfeld, Elco, Elder, Elderton, Eldred, Elgin, Elizabeth, Elizabethtown, Elizabethville, Elk, Elk Creek, Elk Lick, Elkland, Ellport, Ellsworth, Ellwood City, Elmhurst, Elverson, Emlenton, Emmaus, Emporium, Emsworth, Enon Valley, Ephrata, Erie, Ernest, Etna, Eulalia, Evans City, Everett, Everson, Exeter, Export, Factoryville, Fairchance, Fairfield, Fairhope, Fairmount, Fairview, Fallowfield, Falls, Falls Creek, Fallston, Fannett, Farmington, Farrell, Fawn, Fawn Grove, Fayette, Fayette City, Fell, Felton, Ferguson, Fermanagh, Ferndale, Findlay, Findley, Finleyville, Fishing Creek, Fleetwood, Flemington, Folcroft, Ford City, Ford Cliff, Forest City, Forest Hills, Forest Lake, Forks, Forkston, Forksville, Forty Fort, Forward, Foster, Fountain Hill, Fox, Fox Chapel, Foxburg, Frackville, Frailey, Franconia, Frankfort Springs, Franklin, Franklin Park, Franklintown, Frankstown, Frazer, Fredonia, Freeburg, Freedom, Freehold, Freeland, Freemansburg, Freeport, French Creek, Frenchcreek, Friendsville, Fulton, Gaines, Galeton, Gallagher, Gallitzin, Gamble, Garrett, Gaskill, Geistown, Genesee, Georges, Georgetown, German, Germany, Gettysburg, Gibson, Gilberton, Gilmore, Gilpin, Girard, Girardville, Glade, Glasgow, Glassport, Glen Campbell, Glen Hope, Glen Osborne, Glen Rock, Glenburn, Glendon, Glenfield, Glenolden, Goldsboro, Gordon, Goshen, Graham, Grampian, Grant, Granville, Gratz, Gray, Great Bend, Green, Green Hills, Green Lane, Green Tree, Greencastle, Greene, Greenfield, Greensboro, Greensburg, Greenville, Greenwich, Greenwood, Gregg, Grove, Grove City, Grugan, Guilford, Gulich, Haines, Halfmoon, Halifax, Hallam, Hallstead, Hamburg, Hamilton, Hamiltonban, Hamlin, Hampden, Hampton, Hanover, Harborcreek, Harford, Harmar, Harmony, Harris, Harrisburg, Harrison, Harrisville, Hartleton, Hartley, Harveys Lake, Hastings, Hatboro, Hatfield, Haverford, Hawley, Hawthorn, Haycock, Hayfield, Haysville, Hazle, Hazleton, Heath, Hebron, Hector, Hegins, Heidelberg, Hellam, Hellertown, Hemlock, Hempfield, Henderson, Henry Clay, Hepburn, Hereford, Hermitage, Herndon, Herrick, Hickory, Highland, Highspire, Hillsgrove, Hilltown, Hollenback, Hollidaysburg, Homer, Homer City, Homestead, Homewood, Honesdale, Honey Brook, Hookstown, Hooversville, Hop Bottom, Hopewell, Horsham, Horton, Houston, Houtzdale, Hovey, Howard, Howe, Hubley, Hughestown, Hughesville, Hulmeville, Hummelstown, Hunker, Hunlock, Huntingdon, Huntington, Huston, Hyde Park, Hydetown, Hyndman, Independence, Indian Lake, Indiana, Industry, Ingram, Irvona, Irwin, Ivyland, Jackson, Jackson Center, Jacobus, Jamestown, Jay, Jeannette, Jeddo, Jefferson, Jefferson Hills, Jenkins, Jenkintown, Jenks, Jenner, Jennerstown, Jermyn, Jersey Shore, Jessup, Jim Thorpe, Johnsonburg, Johnstown, Jones, Jonestown, Jordan, Juniata, Juniata Terrace, Kane, Karns City, Karthaus, Keating, Kelly, Kenhorst, Kennedy, Kennett, Kennett Square, Kidder, Kilbuck, Kimmel, King, Kingsley, Kingston, Kiskiminetas, Kistler, Kittanning, Kline, Knox, Knoxville, Koppel, Kulpmont, Kutztown, Laceyville, Lack, Lackawannock, Lackawaxen, Lafayette, Laflin, Lake, Lake City, Lamar, Lancaster, Landingville, Landisburg, Lanesboro, Langhorne, Langhorne Manor, Lansdale, Lansdowne, Lansford, Laplume, Laporte, Larimer, Larksville, Lathrop, Latimore, Latrobe, Laurel Mountain, Laurel Run, Laureldale, Lausanne, Lawrence, Lawrence Park, Lawrenceville, Le Boeuf, Leacock, Lebanon, Leechburg, Leesport, Leet, Leetsdale, Lehigh, Lehighton, Lehman, Leidy, Lemon, Lemoyne, Lenhartsville, Lenox, Leraysville, Leroy, Letterkenny, Lewis, Lewis Run, Lewisberry, Lewisburg, Lewistown, Liberty, Licking, Licking Creek, Ligonier, Lilly, Limerick, Limestone, Lincoln, Linesville, Litchfield, Lititz, Little Beaver, Little Britain, Little Mahanoy, Little Meadows, Littlestown, Liverpool, Lock Haven, Locust, Logan, Loganton, Loganville, London Britain, London Grove, Londonderry, Long Branch, Longswamp, Lorain, Loretto, Lower Allen, Lower Alsace, Lower Augusta, Lower Burrell, Lower Chanceford, Lower Chichester, Lower Frankford, Lower Frederick, Lower Gwynedd, Lower Heidelberg, Lower Macungie, Lower Mahanoy, Lower Makefield, Lower Merion, Lower Mifflin, Lower Milford, Lower Moreland, Lower Mount Bethel, Lower Nazareth, Lower Oxford, Lower Paxton, Lower Pottsgrove, Lower Providence, Lower Salford, Lower Saucon, Lower Southampton, Lower Swatara, Lower Towamensing, Lower Turkeyfoot, Lower Tyrone, Lower Windsor, Lower Yoder, Lowhill, Loyalhanna, Loyalsock, Lumber, Lurgan, Luzerne, Lycoming, Lykens, Lynn, Lyons, Macungie, Madison, Mahaffey, Mahanoy, Mahanoy City, Mahoning, Maidencreek, Main, Malvern, Manchester, Manheim, Mann, Manns Choice, Manor, Manorville, Mansfield, Mapleton, Marcus Hook, Marianna, Marietta, Marion, Marion Center, Marion Heights, Marklesburg, Markleysburg, Marlborough, Marple, Mars, Marshall, Martic, Martinsburg, Marysville, Masontown, Matamoras, Maxatawny, Mayberry, Mayfield, Mcadoo, Mccalmont, Mccandless, Mcclure, Mcconnellsburg, Mcdonald, Mcewensville, Mchenry, Mcintyre, Mckean, Mckees Rocks, Mckeesport, Mcnett, Mcsherrystown, Mcveytown, Mead, Meadville, Mechanicsburg, Mechanicsville, Media, Mehoopany, Menallen, Menno, Mercer, Mercersburg, Meshoppen, Metal, Meyersdale, Middle Paxton, Middle Smithfield, Middle Taylor, Middleburg, Middlebury, Middlecreek, Middleport, Middlesex, Middletown, Midland, Midway, Mifflin, Mifflinburg, Mifflintown, Miles, Milesburg, Milford, Mill Creek, Mill Hall, Mill Village, Millbourne, Millcreek, Miller, Millersburg, Millerstown, Millersville, Millheim, Millstone, Millvale, Millville, Milton, Mineral, Minersville, Modena, Mohnton, Monaca, Monaghan, Monessen, Monongahela, Monroe, Monroeville, Mont Alto, Montgomery, Montour, Montoursville, Montrose, Moon, Moore, Moosic, Moreland, Morgan, Morris, Morrisville, Morton, Moscow, Mount Carbon, Mount Carmel, Mount Gretna, Mount Holly Springs, Mount Jewett, Mount Joy, Mount Lebanon, Mount Oliver, Mount Penn, Mount Pleasant, Mount Pocono, Mount Union, Mount Wolf, Mountville, Muddy Creek, Muhlenberg, Muncy, Muncy Creek, Munhall, Munster, Murrysville, Myerstown, Nanticoke, Nanty Glo, Napier, Narberth, Nazareth, Nelson, Nescopeck, Neshannock, Nesquehoning, Nether Providence, Neville, New Albany, New Alexandria, New Baltimore, New Beaver, New Berlin, New Bethlehem, New Brighton, New Britain, New Buffalo, New Castle, New Centerville, New Columbus, New Cumberland, New Eagle, New Florence, New Freedom, New Galilee, New Garden, New Hanover, New Holland, New Hope, New Kensington, New Lebanon, New London, New Milford, New Morgan, New Oxford, New Paris, New Philadelphia, New Ringgold, New Salem, New Sewickley, New Stanton, New Vernon, New Washington, New Wilmington, Newberry, Newburg, Newell, Newlin, Newport, Newry, Newton, Newton Hamilton, Newtown, Newville, Nicholson, Nippenose, Nockamixon, Norristown, North Abington, North Annville, North Apollo, North Beaver, North Belle Vernon, North Bethlehem, North Braddock, North Branch, North Buffalo, North Catasauqua, North Centre, North Charleroi, North Codorus, North Cornwall, North Coventry, North East, North Fayette, North Franklin, North Heidelberg, North Hopewell, North Huntingdon, North Irwin, North Lebanon, North Londonderry, North Mahoning, North Manheim, North Middleton, North Newton, North Sewickley, North Shenango, North Strabane, North Towanda, North Union, North Versailles, North Wales, North Whitehall, North Woodbury, North York, Northampton, Northeast Madison, Northern Cambria, Northmoreland, Northumberland, Norwegian, Norwich, Norwood, Nottingham, Noxen, Noyes, Nuangola, Oakdale, Oakland, Oakmont, Ogle, Ohara, Ohio, Ohiopyle, Ohioville, Oil City, Oil Creek, Oilcreek, Oklahoma, Old Forge, Old Lycoming, Oley, Oliver, Olyphant, Oneida, Ontelaunee, Orange, Orangeville, Orbisonia, Oregon, Orrstown, Orwell, Orwigsburg, Osceola, Osceola Mills, Oswayo, Otter Creek, Otto, Overfield, Overton, Oxford, Packer, Paint, Palmer, Palmerton, Palmyra, Palo Alto, Paradise, Parker, Parkesburg, Parks, Parkside, Parryville, Patterson, Patterson Heights, Patton, Paupack, Pavia, Paxtang, Peach Bottom, Pen Argyl, Penbrook, Penn, Penn Forest, Penn Hills, Penn Lake Park, Penndel, Pennsburg, Pennsbury, Pennsbury Village, Pequea, Perkasie, Perkiomen, Perry, Perryopolis, Peters, Petersburg, Petrolia, Philadelphia, Philipsburg, Phoenixville, Piatt, Picture Rocks, Pike, Pillow, Pine, Pine Creek, Pine Grove, Pinegrove, Piney, Pitcairn, Pittsburgh, Pittsfield, Pittston, Plain Grove, Plainfield, Plains, Platea, Pleasant, Pleasant Hills, Pleasant Valley, Pleasantville, Plum, Plumcreek, Plumstead, Plumville, Plunketts Creek, Plymouth, Pocono, Pocopson, Point, Point Marion, Polk, Port Allegany, Port Carbon, Port Clinton, Port Matilda, Port Royal, Port Vue, Portage, Porter, Portersville, Portland, Potter, Pottstown, Pottsville, President, Preston, Price, Pringle, Prompton, Prospect, Prospect Park, Providence, Pulaski, Punxsutawney, Putnam, Pymatuning, Quakertown, Quarryville, Quemahoning, Quincy, Raccoon, Radnor, Railroad, Rainsburg, Ralpho, Ramey, Randolph, Rankin, Ransom, Rapho, Rayburn, Rayne, Reade, Reading, Red Hill, Red Lion, Redbank, Redstone, Reed, Reilly, Renovo, Reserve, Reynoldsville, Rice, Rices Landing, Richhill, Richland, Richlandtown, Richmond, Ridgebury, Ridgway, Ridley, Ridley Park, Riegelsville, Rimersburg, Ringgold, Ringtown, Riverside, Roaring Brook, Roaring Creek, Roaring Spring, Robeson, Robesonia, Robinson, Rochester, Rockdale, Rockefeller, Rockhill, Rockland, Rockledge, Rockwood, Rome, Roscoe, Rose, Rose Valley, Roseto, Roseville, Ross, Rosslyn Farms, Rostraver, Roulette, Rouseville, Royalton, Royersford, Rural Valley, Ruscombmanor, Rush, Rutland, Rutledge, Ryan, Rye, S.N.P.J., Sadsbury, Saegertown, Salem, Salford, Salisbury, Salladasburg, Saltillo, Saltlick, Saltsburg, Sandy, Sandy Creek, Sandy Lake, Sandycreek, Sankertown, Saville, Saxonburg, Saxton, Sayre, Scalp Level, Schellsburg, Schuylkill, Schuylkill Haven, Schwenksville, Scott, Scottdale, Scranton, Scrubgrass, Selinsgrove, Sellersville, Sergeant, Seven Fields, Seven Springs, Seven Valleys, Seward, Sewickley, Sewickley Heights, Sewickley Hills, Shade, Shade Gap, Shaler, Shamokin, Shamokin Dam, Shanksville, Sharon, Sharon Hill, Sharpsburg, Sharpsville, Sheakleyville, Sheffield, Shelocta, Shenandoah, Shenango, Sheshequin, Shickshinny, Shillington, Shinglehouse, Shippen, Shippensburg, Shippenville, Shippingport, Shiremanstown, Shirley, Shirleysburg, Shoemakersville, Shohola, Shrewsbury, Silver Lake, Silver Spring, Silverdale, Sinking Spring, Skippack, Slatington, Sligo, Slippery Rock, Slocum, Smethport, Smicksburg, Smith, Smithfield, Smithton, Snake Spring, Snow Shoe, Snyder, Snydertown, Solebury, Somerset, Souderton, South Abington, South Annville, South Beaver, South Bend, South Bethlehem, South Buffalo, South Canaan, South Centre, South Coatesville, South Connellsville, South Coventry, South Creek, South Fayette, South Fork, South Franklin, South Greensburg, South Hanover, South Heidelberg, South Heights, South Huntingdon, South Lebanon, South Londonderry, South Mahoning, South Manheim, South Middleton, South New Castle, South Newton, South Park, South Pymatuning, South Renovo, South Shenango, South Strabane, South Union, South Versailles, South Waverly, South Whitehall, South Williamsport, South Woodbury, Southampton, Southmont, Southwest, Southwest Greensburg, Southwest Madison, Sparta, Spartansburg, Speers, Spring, Spring Brook, Spring City, Spring Creek, Spring Garden, Spring Grove, Springboro, Springdale, Springettsbury, Springfield, Springhill, Springville, Spruce Creek, Spruce Hill, St Clair, St Clairsville, St Lawrence, St Marys, St Petersburg, St Thomas, Standing Stone, Starrucca, State College, Steelton, Sterling, Steuben, Stevens, Stewardson, Stewart, Stewartstown, Stillwater, Stockdale, Stockertown, Stoneboro, Stonycreek, Stowe, Stoystown, Straban, Strasburg, Strattanville, Stroud, Stroudsburg, Sugar Grove, Sugar Notch, Sugarcreek, Sugarloaf, Sullivan, Summerhill, Summerville, Summit, Summit Hill, Sunbury, Susquehanna, Susquehanna Depot, Sutersville, Swarthmore, Swatara, Sweden, Swissvale, Swoyersville, Sykesville, Sylvania, Tamaqua, Tarentum, Tatamy, Taylor, Telford, Tell, Terre Hill, Terry, Texas, Thompson, Thompsontown, Thornburg, Thornbury, Thornhurst, Three Springs, Throop, Tidioute, Tilden, Timblin, Tinicum, Tioga, Tionesta, Titusville, Toboyne, Toby, Tobyhanna, Todd, Topton, Towamencin, Towamensing, Towanda, Tower City, Townville, Trafford, Trainer, Trappe, Tredyffrin, Tremont, Triumph, Troutville, Troy, Trumbauersville, Tullytown, Tulpehocken, Tunkhannock, Tunnelhill, Turbett, Turbot, Turbotville, Turtle Creek, Tuscarora, Twilight, Tyrone, Ulster, Ulysses, Union, Union City, Uniondale, Uniontown, Unionville, Unity, Upland, Upper Allen, Upper Augusta, Upper Bern, Upper Burrell, Upper Chichester, Upper Darby, Upper Dublin, Upper Fairfield, Upper Frankford, Upper Frederick, Upper Gwynedd, Upper Hanover, Upper Leacock, Upper Macungie, Upper Mahanoy, Upper Mahantango, Upper Makefield, Upper Merion, Upper Mifflin, Upper Milford, Upper Moreland, Upper Mount Bethel, Upper Nazareth, Upper Oxford, Upper Paxton, Upper Pottsgrove, Upper Providence, Upper Salford, Upper Saucon, Upper Southampton, Upper St Clair, Upper Tulpehocken, Upper Turkeyfoot, Upper Tyrone, Upper Uwchlan, Upper Yoder, Ursina, Utica, Uwchlan, Valencia, Valley, Valley-Hi, Vanderbilt, Vandergrift, Vandling, Vanport, Venango, Vernon, Verona, Versailles, Victory, Vintondale, Volant, Walker, Wall, Wallace, Wallaceton, Walnutport, Wampum, Ward, Warminster, Warren, Warrington, Warrior Run, Warriors Mark, Warsaw, Warwick, Washington, Washingtonville, Waterford, Watson, Watsontown, Watts, Wattsburg, Waverly, Waymart, Wayne, Waynesboro, Waynesburg, Weatherly, Weisenberg, Weissport, Wellersburg, Wells, Wellsboro, Wellsville, Wernersville, Wesleyville, West, West Abington, West Beaver, West Bethlehem, West Bradford, West Branch, West Brandywine, West Brownsville, West Brunswick, West Buffalo, West Burlington, West Caln, West Cameron, West Carroll, West Chester, West Chillisquaque, West Cocalico, West Conshohocken, West Cornwall, West Deer, West Donegal, West Earl, West Easton, West Elizabeth, West Fallowfield, West Finley, West Franklin, West Goshen, West Grove, West Hanover, West Hazleton, West Hemlock, West Hempfield, West Homestead, West Keating, West Kittanning, West Lampeter, West Lebanon, West Leechburg, West Liberty, West Mahanoy, West Mahoning, West Manchester, West Manheim, West Marlborough, West Mayfield, West Mead, West Middlesex, West Middletown, West Mifflin, West Nantmeal, West Newton, West Norriton, West Nottingham, West Penn, West Pennsboro, West Perry, West Pike Run, West Pikeland, West Pittston, West Pottsgrove, West Providence, West Reading, West Rockhill, West Sadsbury, West Salem, West Shenango, West St Clair, West Sunbury, West Taylor, West View, West Vincent, West Wheatfield, West Whiteland, West Wyoming, West York, Westfall, Westfield, Westmont, Westover, Westtown, Wetmore, Wharton, Wheatfield, Wheatland, Whitaker, White, White Deer, White Haven, White Oak, Whitehall, Whiteley, Whitemarsh, Whitpain, Wiconisco, Wilkes Barre, Wilkins, Wilkinsburg, Williams, Williamsburg, Williamsport, Williamstown, Willistown, Wilmerding, Wilmington, Wilmore, Wilmot, Wilson, Wind Gap, Windber, Windham, Windsor, Winfield, Winslow, Winterstown, Wolf, Wolf Creek, Womelsdorf, Wood, Woodbury, Woodcock, Woodward, Worcester, Wormleysburg, Worth, Worthington, Worthville, Wright, Wrightstown, Wrightsville, Wyalusing, Wyoming, Wyomissing, Wysox, Yardley, Yatesville, Yeadon, Yoe, York, York Haven, York Springs, Yorkana, Young, Youngstown, Youngsville, Youngwood, Zelienople, Zerbe

next, we turn our attention to Census Designated Places from https://www2.census.gov/geo/docs/maps-data/data/gazetteer/2021_Gazetteer/2021_gaz_place_42.txt.

the Census list has CDP but not townships.  they overlap among boroughs, cities, and town.

Bethel Park, Monroeville, Murrysville are "Municipality" not Borough.

CamelCase for DuBois, East McKeesport, LeRaysville, McAdoo, McClure, McConnellsburg, McDonald, McEwensville, McKean, McKees Rocks, McKeesport, McSherrystown, McVeytown.

period consistently after "St" for St. Clair, St. Clairsville, St. Lawrence, St. Marys, St. Petersburg.

"Mt" consistently expanded to "Mount".

hyphenation for Nanty-Glo and Wilkes-Barre.

it has Union Dale not Uniondale.

CDPs whose names contain punctuation: Bird-in-Hand, Brittany Farms-The Highlands, Carnot-Moon, Cecil-Bishop, Fairview-Ferndale, Homeacre-Lyndora, Leith-Hatfield, Lynnwood-Pricedale, Penn State Erie (Behrend), Shanor-Northvue, St. Clair, St. Davids, St. Michael, St. Vincent College, Tharptown (Uniontown), Village Green-Green Ridge, Wickerham Manor-Fisher

also of interest: Wilburton Number One, Wilburton Number Two

here are the "Name:Type" that occur more than once: "Aaronsburg:CDP" (2), "Centerville:Borough" (2), "Coaldale:Borough" (2), "Eagleville:CDP" (2), "Georgetown:CDP" (2), "Jacksonville:CDP" (2), "Jefferson:Borough" (2), "Liberty:Borough" (2), "Marshallton:CDP" (2), "Newburg:Borough" (2), "Oakland:CDP" (2), "Pleasantville:Borough" (2), "Unionville:CDP" (2), "Valley View:CDP" (2), "Vicksburg:CDP" (2), "Westwood:CDP" (2), "Fredericksburg:CDP" (3),

Chestnut Hill and Spring Hill are CDP; Chestnuthill and Springhill are townships.

after eliminating CDP names that overlap with township names, below are 778 more places.  this list includes some of the more famous unusual Pennsylvania place names: King of Prussia, Eighty Four.

Aaronsburg, Ackermanville, Adamsville, Albrightsville, Alfarata, Alleghenyville, Allensville, Allenwood, Allison Park, Allport, Almedia, Alsace Manor, Altamont, Alverda, Amity Gardens, Ancient Oaks, Arcadia University, Ardmore, Aristes, Arlington Heights, Arnold City, Arnot, Aspers, Atkinson Mills, Atlantic, Atlas, Atlasburg, Audubon, Avella, Avon, Avonia, Baidland, Baileyville, Bainbridge, Bairdford, Bakerstown, Bala Cynwyd, Barrville, Baumstown, Bear Rocks, Beaver Springs, Beaverdale, Beech Mountain Lakes, Belleville, Belmont, Berwyn, Beurys Lake, Big Bass Lake, Birchwood Lakes, Bird-in-Hand, Blanchard, Blandburg, Blandon, Blue Ball, Blue Bell, Blue Knob, Blue Ridge Summit, Boalsburg, Bobtown, Boiling Springs, Boothwyn, Boston, Bowers, Bowmansville, Bradenville, Branchdale, Brandonville, Brave, Breinigsville, Bressler, Brickerville, Brittany Farms-The Highlands, Brodheadsville, Brooks Mill, Broomall, Browntown, Bryn Mawr, Buck Run, Buckhorn, Bulger, Byrnedale, Cairnbrook, Calumet, Campbelltown, Canadohta Lake, Canan Station, Canoe Creek, Carlisle Barracks, Carnot-Moon, Carson Valley, Casanova, Cashtown, Cecil-Bishop, Cedar Crest, Cementon, Cetronia, Chalkhill, Charlottsville, Chase, Cheltenham Village, Cherryville, Chesterbrook, Chestnut Hill, Chevy Chase Heights, Chewton, Cheyney University, Chinchilla, Church Hill, Churchtown, Churchville, Clappertown, Clarence, Claysburg, Clintondale, Coburn, Cochranville, Collinsburg, Colonial Park, Colony Park, Colver, Commodore, Conashaugh Lakes, Conneaut Lakeshore, Continental Courts, Coral, Cornwells Heights, Cotton Town, Cove Forge, Crabtree, Crenshaw, Cross Keys, Crown, Croydon, Crucible, Culp, Cumbola, Curryville, Curtisville, Dalmatia, Dauberville, Davidsville, DeSales University, Defiance, Devon, Dewart, Dilworthtown, Dixonville, Donaldson, Dorneyville, Douglassville, Drexel Hill, Dry Tavern, Dryville, Dumb Hundred, Dunlo, Dunnstown, Eagle Creek, Eagle Lake, Eagle, Eagleview, Eagleville, Earlston, East Altoona, East Berwick, East Freedom, East Salem, East Sharpsburg, East Uniontown, East Waterford, East York, Eastlawn Gardens, Eddington, Edenborn, Edenburg, Edie, Effort, Egypt, Eighty Four, Elberta, Eldorado, Elim, Elkins Park, Elrama, Elysburg, Emerald Lakes, Emigsville, Englewood, Enhaut, Enlow, Enola, Espy, Evansburg, Exton, Eyers Grove, Fairdale, Fairless Hills, Fairview-Ferndale, Falmouth, Farmersville, Farwell, Fawn Lake Forest, Faxon, Fayetteville, Feasterville, Fellsburg, Fernville, Fisherville, Fivepointville, Flourtown, Flying Hills, Folsom, Foot of Ten, Force, Forestville, Fort Fetter, Fort Indiantown Gap, Fort Loudon, Fort Washington, Foster Brook, Fostoria, Foundryville, Fountain Springs, Fox Chase, Franklin Forge, Fredericksburg, Fredericktown, Friedens, Friedensburg, Friesville, Frisco, Frizzleburg, Frystown, Fullerton, Ganister, Gap, Garden View, Gardners, Gastonville, Geeseytown, Geneva, Gibraltar, Gibsonia, Gilbertsville, Glen Lyon, Glenmoore, Glenshaw, Glenside, Gold Key Lake, Goodville, Gordonville, Gouglersville, Gouldsboro, Graceton, Grantley, Grapeville, Grassflat, Grazierville, Greenfields, Greenock, Greens Landing, Grier City, Grill, Grindstone, Guys Mills, Halfway House, Hamorton, Hannasville, Harleigh, Harleysville, Harmonsburg, Harrison City, Hartstown, Harwick, Hasson Heights, Haverford College, Hawk Run, Hayti, Hazen, Heckscherville, Heidlersburg, Heilwood, Hemlock Farms, Hendersonville, Henrietta, Herminie, Hershey, Hickory Hills, Highland Park, Hilldale, Hiller, Hokendauqua, Holiday Pocono, Holters Crossing, Homeacre-Lyndora, Hometown, Homewood at Martinsburg, Hopeland, Hopwood, Hostetter, Houserville, Hublersburg, Hudson, Hummels Wharf, Hunterstown, Hyde, Idaville, Imperial, Indian Mountain Lake, Inkerman, Intercourse, Iola, Ironville, Jacksonville, Jacksonwald, James City, Jamison City, Jerome, Jerseytown, Joffre, Jugtown, Julian, Juniata Gap, Kapp Heights, Kelayres, Kempton, Kenilworth, Kenmar, Kennerdell, Kerrtown, Kersey, Kimberton, King of Prussia, Kirkwood, Kiskimere, Klahr, Klingerstown, Kratzerville, Kreamer, Kulpsville, Kutztown University, Kylertown, Lake Arthur Estates, Lake Heritage, Lake Latonka, Lake Meade, Lake Winola, Lake Wynonah, Lakemont, Lampeter, Landisville, Langeloth, Larke, Lattimer, Laurelton, Laurys Station, Lavelle, Lawnton, Lawson Heights, Leamersville, Lebanon South, Leeper, Leith-Hatfield, Lemont Furnace, Lemont, Lenape Heights, Lenkerville, Leola, Level Green, Levittown, Lightstreet, Lima, Lime Ridge, Lincoln Park, Lincoln University, Lincolnville, Linds Crossing, Linglestown, Linntown, Linwood, Lionville, Locustdale, Longfellow, Loop, Lorane, Lucerne Mines, Lumber City, Lynnwood-Pricedale, Madisonburg, Mainville, Maitland, Mammoth, Maple Glen, Mapletown, Marianne, Marienville, Marlin, Marshallton, Martins Creek, Martinsburg Junction, Masthope, Mather, Mattawana, Maytown, McAlisterville, McConnellstown, McElhattan, McGovern, McKeansburg, McKee, McKinley, McKnightstown, McMurray, Meadowlands, Meadowood, Meridian, Merion Station, Mertztown, Messiah College, Mexico, Mifflinville, Milford Square, Mill Run, Millerton, Millsboro, Millwood, Milroy, Mingoville, Misericordia University, Mocanaqua, Mohrsville, Mont Clare, Montandon, Montgomeryville, Montrose Manor, Monument, Moores Mill, Moose Run, Morea, Morgan Hill, Morgantown, Morrisdale, Moshannon, Mount Aetna, Mount Cobb, Mount Eagle, Mount Gretna Heights, Mount Morris, Mount Pleasant Mills, Mountain Top, Mountainhome, Muhlenberg Park, Muir, Mundys Corner, Muse, Naomi, Nealmont, Needmore, Nemacolin, New Bedford, New Berlinville, New Boston, New Castle Northwest, New Columbia, New Freeport, New Jerusalem, New Kingstown, New Market, New Schaefferstown, New Tripoli, Newmanstown, Newtown Grant, Nittany, Nixon, Noblestown, North Bend, North Philipsburg, North Vandergrift, North Warren, Northwest Harborcreek, Northwood, Norvelt, Numidia, Nuremberg, Oak Hills, Oakwood, Oberlin, Old Orchard, Olivia, Orchard Hills, Ore Hill, Oreland, Oreminea, Orrtanna, Orviston, Orwin, Oval, Palmdale, Palmer Heights, Paoli, Pardeesville, Paris, Park Crest, Park Forest Village, Parkville, Paxtonia, Paxtonville, Pen Mar, Penn Estates, Penn Farms, Penn State Berks, Penn State Erie (Behrend), Penn Wynne, Penns Creek, Pennside, Pennville, Pennwyn, Penryn, Peru, Pikes Creek, Pine Glen, Pine Grove Mills, Pine Ridge, Pinecroft, Pleasant Gap, Pleasant Hill, Pleasant View, Pleasureville, Plumsteadville, Plymouth Meeting, Plymptonville, Pocono Mountain Lake Estates, Pocono Pines, Pocono Ranch Lands, Pocono Springs, Pocono Woodland Lakes, Point View, Pomeroy, Port Trevorton, Potlicker Flats, Potters Mills, Pottsgrove, Progress, Pughtown, Puzzletown, Pymatuning Central, Pymatuning North, Pymatuning South, Queens Gate, Quentin, Ramblewood, Ranshaw, Raubsville, Rauchtown, Ravine, Reamstown, Rebersburg, Reedsville, Reese, Refton, Rehrersburg, Reiffton, Reightown, Reinerton, Reinholds, Rennerdale, Renningers, Republic, Reservoir, Revloc, Rew, Reynolds Heights, Rheems, Riceville, Richboro, Richfield, Riverview Park, Robeson Extension, Rogersville, Rohrsburg, Ronco, Ronks, Roots, Rosemont, Roslyn, Rossiter, Rote, Rothsville, Rouzerville, Rowes Run, Royer, Runville, Rupert, Russell, Russellton, Rutherford, Sadsburyville, Salix, Salona, Salunga, Sanatoga, Sand Hill, Sandy Ridge, Saw Creek, Saylorsburg, Schaefferstown, Schlusser, Schnecksville, Schoeneck, Schubert, Scotland, Seltzer, Seneca, Shanor-Northvue, Shartlesville, Shavertown, Sheatown, Shelltown, Shellytown, Shenandoah Heights, Sheppton, Shiloh, Shippensburg University, Sickles Corner, Sidman, Sierra View, Siglerville, Silkworth, Simpson, Skelp, Ski Gap, Skyline View, Slabtown, Slatedale, Slickville, Slippery Rock University, Slovan, Smith Corner, Smock, Smoketown, Soudersburg, South Philipsburg, South Pottstown, South Temple, South Uniontown, Southview, Spinnerstown, Spring Drive Mobile Home Park, Spring Hill, Spring House, Spring Mills, Spring Mount, Spring Ridge, Springmont, Sproul, Spry, St. Davids, St. Michael, St. Vincent College, Star Junction, Starbrick, State Line, Stiles, Stonerstown, Stony Creek Mills, Stonybrook, Stormstown, Stouchsburg, Strausstown, Strodes Mills, Strong, Sturgeon, Summit Station, Sun Valley, Sunbrook, Sunrise Lake, Susquehanna Trails, Swartzville, Swedeland, Sweden Valley, Sylvan Hills, Table Rock, Taylorstown, Temple, Templeton, Tharptown (Uniontown), The Hideout, Thompsonville, Thorndale, Timber Hills, Tipton, Toftrees, Toughkenamon, Towamensing Trails, Treasure Lake, Tresckow, Trevorton, Trevose, Trexlertown, Trooper, Troxelville, Trucksville, Tyler Run, Tylersburg, Tylersville, Tyrone Forge, Union Deposit, University of Pittsburgh Bradford, University of Pittsburgh Johnstown, Upper Exeter, Vail, Valley Green, Valley View, Van Voorhis, Vicksburg, Village Green-Green Ridge, Village Shires, Villanova, Vinco, Virginville, Vowinckel, Wagner, Wakefield, Wallenpaupack Lake Estates, Waller, Walnuttown, Wanamie, Warminster Heights, Washington Boro, Wayne Heights, Webster, Weedville, Weigelstown, Weissport East, Wescosville, West Alexander, West Decatur, West Fairview, West Falls, West Hamburg, West Hills, West Lawn, West Milton, West Nanticoke, West Pittsburg, West Waynesburg, West Wyomissing, Westland, Weston, Westwood, White Mills, Whitfield, Wickerham Manor-Fisher, Wiconsico, Wilburton Number One, Wilburton Number Two, Wilcox, Willow Grove, Willow Street, Wind Ridge, Witmer, Wolfdale, Woodbourne, Woodland Heights, Woodlyn, Woodside, Woolrich, Wopsononock, Woxall, Wyano, Wylandville, Wyncote, Wyndmoor, Yarnell, Yeagertown, Yellow Springs, Yorklyn, Yukon, Zion

county names (6 out of 67 counties total) which don't overlap with the above two lists: Berks, Bucks, Cameron, Forest, Lackawanna, Westmoreland

future work: the USGS Geographic Names Information System has even more names.  of particular interest is Census Class Code "U6" (unincorporated communities that are not census designated places) from "Individual State Files with Federal Codes", which contains some 6000 more places.  the online interface may be useful for understanding the data.

although some of the names are not very creative, the list represents a distributed process of human creativity over hundreds of years.  someone had to come up with each name.

[tvpthwpj] primitive binary pentanomials

primitive binary trinomials, i.e., primitive polynomials over the field GF(2) with three terms including a nonzero constant term (i.e., 1), only exist for some orders (by order we mean the largest exponent).  however, 5-term polynomials seem to always exist (not proven).  we list the lexicographically smallest ones below, giving the exponents of the middle terms.  for example, the entry "8: 4 3 2" means x^8 + x^4 + x^3 + x^2 + 1 (mod 2) is a primitive polynomial, and among all x^8 + x^A + x^B + x^C + 1, it minimizes A first, then B, then C.

there are no primitive binary tetranomials (4 terms).  proof by contradiction: assume one exists.  evaluate it at x=1.  because there are 4 terms, the answer is 4, which is 0 (mod 2), so x=1 is a root of the hypothetical polynomial.  the polynomial will therefore divide through evenly by (x-1), equivalently (x+1) because +1 = -1 (mod 2), meaning it is reducible, not irreducible.  if it is reducible, it cannot be primitive (though I don't understand why).

order 5 has the smallest binary primitive pentanomial: x^5 + x^3 + x^2 + x + 1

the order 4 pentanomial x^4 + x^3 + x^2 + x + 1 is irreducible but not primitive, because x^5 (mod x^4 + x^3 + x^2 + x + 1) = 1, or x^5 + 1 = (x^4 + x^3 + x^2 + x + 1) * (x + 1).  x^5 is less than x^15, the full period.

proving a polynomial with highest exponent N primitive requires the prime factorization of the number 2^N-1.  with some exceptions, these have been computed by the Cunningham project up to 1499.  the orders marked UNFACTORED below represent the state as of April 2022.

below is simplified Paris/GP code to find the lexicographically smallest pentanomial for a given order.  it is simplified in that it does not look up UNFACTORED status.

penta(n)= my(cy=2^n-1); my(ff=factorint(cy)); nf=matsize(ff)[1]; for(k1=1, n-1, for(k2=1, k1-1, for(k3=1, k2-1, p=Mod(1,2)*x^n+x^k1+x^k2+x^k3+1; if(polisirreducible(p), g=Mod(x,p); good=1; for(j=1, nf, eo=cy/ff[j,1]; raised=g^eo; if(raised==1, good=0; break)); if(good, print(n": "k1" "k2" "k3); return(1)))))); 0;

5: 3 2 1
6: 4 3 1
7: 3 2 1
8: 4 3 2
9: 4 3 1
10: 4 3 1
11: 4 2 1
12: 6 4 1
13: 4 3 1
14: 5 3 1
15: 4 2 1
16: 5 3 2
17: 3 2 1
18: 5 2 1
19: 5 2 1
20: 6 4 1
21: 5 2 1
22: 5 4 3
23: 5 3 1
24: 4 3 1
25: 3 2 1
26: 6 2 1
27: 5 2 1
28: 6 4 1
29: 4 2 1
30: 6 4 1
31: 3 2 1
32: 7 6 2
33: 6 4 1
34: 8 4 3
35: 8 7 1
36: 8 7 1
37: 6 4 1
38: 6 5 1
39: 7 4 1
40: 5 4 3
41: 3 2 1
42: 7 4 3
43: 6 4 3
44: 6 5 2
45: 4 3 1
46: 8 7 6
47: 5 4 1
48: 9 7 4
49: 6 5 4
50: 4 3 2
51: 6 3 1
52: 6 3 1
53: 6 2 1
54: 8 6 3
55: 6 2 1
56: 7 4 2
57: 5 3 2
58: 6 5 1
59: 7 4 2
60: 5 4 2
61: 5 2 1
62: 6 5 3
63: 5 4 1
64: 4 3 1
65: 4 3 1
66: 9 8 6
67: 5 2 1
68: 7 5 1
69: 6 5 2
70: 5 3 1
71: 5 3 1
72: 10 9 3
73: 4 3 2
74: 7 4 3
75: 6 3 1
76: 5 4 2
77: 6 5 2
78: 7 2 1
79: 4 3 2
80: 9 4 2
81: 6 3 2
82: 9 6 4
83: 7 4 2
84: 9 7 1
85: 8 2 1
86: 6 5 2
87: 7 5 1
88: 11 9 8
89: 6 5 3
90: 5 3 2
91: 8 5 1
92: 6 5 2
93: 6 3 2
94: 6 5 1
95: 7 5 1
96: 10 9 6
97: 6 4 2
98: 8 7 1
99: 7 5 4
100: 8 7 2
101: 7 6 1
102: 6 5 3
103: 9 4 1
104: 11 10 1
105: 7 5 2
106: 6 5 1
107: 9 7 4
108: 12 11 5
109: 5 4 2
110: 6 4 1
111: 7 4 2
112: 11 6 4
113: 5 3 2
114: 11 2 1
115: 8 7 5
116: 6 5 2
117: 5 2 1
118: 6 5 2
119: 9 7 4
120: 9 6 2
121: 8 5 1
122: 6 2 1
123: 8 4 1
124: 7 6 5
125: 7 6 5
126: 7 4 2
127: 7 3 1
128: 7 2 1
129: 5 4 1
130: 5 2 1
131: 8 3 2
132: 9 5 2
133: 9 8 2
134: 7 5 1
135: 6 4 3
136: 8 3 2
137: 11 4 1
138: 8 7 1
139: 8 5 3
140: 8 4 1
141: 13 6 1
142: 10 3 1
143: 5 3 2
144: 7 4 2
145: 6 5 1
146: 5 3 2
147: 11 4 2
148: 7 5 3
149: 10 9 7
150: 8 3 2
151: 3 2 1
152: 6 3 2
153: 8 5 4
154: 9 5 1
155: 7 5 4
156: 9 5 3
157: 6 5 2
158: 8 6 5
159: 11 6 3
160: 5 3 2
161: 6 3 2
162: 8 7 4
163: 7 6 3
164: 12 6 5
165: 9 8 3
166: 10 3 2
167: 6 4 2
168: 16 9 6
169: 8 6 5
170: 9 4 1
171: 6 5 2
172: 11 7 3
173: 8 5 2
174: 9 8 5
175: 6 4 2
176: 12 11 9
177: 5 3 2
178: 8 7 2
179: 4 2 1
180: 12 10 7
181: 7 6 1
182: 8 6 1
183: 8 7 4
184: 9 8 7
185: 8 3 1
186: 9 8 6
187: 7 6 5
188: 6 5 2
189: 6 5 2
190: 13 6 2
191: 7 6 4
192: 15 11 5
193: 9 7 4
194: 4 3 2
195: 8 3 2
196: 11 9 2
197: 9 4 2
198: 15 8 5
199: 9 4 1
200: 5 3 2
201: 6 3 2
202: 7 6 4
203: 8 7 1
204: 10 4 3
205: 9 5 2
206: 10 9 5
207: 9 6 1
208: 9 3 1
209: 5 3 2
210: 12 4 3
211: 11 10 8
212: 7 4 3
213: 6 5 2
214: 5 3 1
215: 6 5 3
216: 7 3 1
217: 6 5 4
218: 8 7 1
219: 8 4 1
220: 12 10 9
221: 8 6 2
222: 8 5 2
223: 5 4 2
224: 12 7 2
225: 10 5 1
226: 10 7 3
227: 10 9 4
228: 12 11 2
229: 10 4 1
230: 8 7 6
231: 7 4 2
232: 11 9 4
233: 9 4 1
234: 11 7 4
235: 9 6 1
236: 10 8 7
237: 7 4 1
238: 5 2 1
239: 12 7 1
240: 8 5 3
241: 9 8 4
242: 11 6 1
243: 8 5 1
244: 9 4 1
245: 6 4 1
246: 11 2 1
247: 9 4 2
248: 15 14 10
249: 7 4 1
250: 10 5 3
251: 7 4 2
252: 11 5 1
253: 7 3 2
254: 7 2 1
255: 5 3 2
256: 10 5 2
257: 7 6 2
258: 9 6 4
259: 10 6 2
260: 10 8 7
261: 7 6 4
262: 9 8 4
263: 11 5 2
264: 10 9 1
265: 5 3 2
266: 7 6 1
267: 8 6 3
268: 10 4 1
269: 7 6 1
270: 10 7 3
271: 11 7 6
272: 9 6 2
273: 7 2 1
274: 9 7 2
275: 11 10 9
276: 6 3 1
277: 12 6 3
278: 5 4 1
279: 5 4 1
280: 9 5 2
281: 9 4 1
282: 10 5 4
283: 12 7 5
284: 8 6 5
285: 10 7 5
286: 15 10 1
287: 6 5 2
288: 11 10 1
289: 12 4 3
290: 5 3 2
291: 12 11 5
292: 7 3 1
293: 11 6 1
294: 9 3 2
295: 5 4 2
296: 11 9 4
297: 5 4 1
298: 11 8 4
299: 11 6 4
300: 13 12 10
301: 9 5 2
302: 12 9 5
303: 13 12 6
304: 11 2 1
305: 7 6 2
306: 7 3 1
307: 8 4 2
308: 15 9 2
309: 10 6 4
310: 8 5 1
311: 7 5 3
312: 11 10 5
313: 7 3 1
314: 14 9 3
315: 10 9 1
316: 12 11 7
317: 7 4 2
318: 8 6 5
319: 11 2 1
320: 4 3 1
321: 7 5 2
322: 17 2 1
323: 10 3 1
324: 6 4 3
325: 10 5 2
326: 10 3 1
327: 8 5 2
328: 9 7 5
329: 8 6 3
330: 8 7 2
331: 10 6 2
332: 12 11 7
333: 8 4 2
334: 7 4 1
335: 10 7 2
336: 7 4 1
337: 10 6 1
338: 6 3 2
339: 16 10 7
340: 11 4 3
341: 14 11 5
342: 11 2 1
343: 10 8 5
344: 11 10 6
345: 8 4 2
346: 11 7 2
347: 11 10 3
348: 8 7 4
349: 6 5 2
350: 14 13 10
351: 8 6 3
352: 13 11 6
353: 9 7 4
354: 14 13 5
355: 6 5 1
356: 10 9 7
357: 11 10 2
358: 14 8 7
359: 9 7 1
360: 26 25 1
361: 7 4 1
362: 18 8 7
363: 8 5 3
364: 12 5 1
365: 9 6 5
366: 14 7 4
367: 9 4 2
368: 17 9 7
369: 11 10 2
370: 5 3 2
371: 8 3 2
372: 15 7 3
373: 8 7 2
374: 8 6 5
375: 8 7 1
376: 8 7 5
377: 8 3 1
378: 15 13 4
379: 10 8 5
380: 14 6 3
381: 5 2 1
382: 18 7 3
383: 9 5 1
384: 16 15 6
385: 6 4 2
386: 10 6 5
387: 9 8 2
388: 14 3 1
389: 10 9 5
390: 13 10 2
391: 6 2 1
392: 13 10 6
393: 7 2 1
394: 8 7 2
395: 11 6 5
396: 7 6 4
397: 12 7 6
398: 14 6 5
399: 11 9 2
400: 5 3 2
401: 12 8 5
402: 9 4 3
403: 9 8 5
404: 7 6 4
405: 17 8 7
406: 13 9 4
407: 9 7 5
408: 7 5 1
409: 7 5 3
410: 10 4 3
411: 12 10 3
412: 11 8 3
413: 10 7 6
414: 16 13 9
415: 9 4 2
416: 9 5 2
417: 10 3 1
418: 15 3 1
419: 15 5 4
420: 13 10 8
421: 5 4 2
422: 10 6 1
423: 9 5 3
424: 9 7 2
425: 7 4 3
426: 14 12 11
427: 11 6 5
428: 11 3 2
429: 10 8 7
430: 15 13 11
431: 5 3 1
432: 13 4 3
433: 11 5 3
434: 12 11 5
435: 12 9 5
436: 6 5 4
437: 6 2 1
438: 17 6 2
439: 8 3 2
440: 4 3 1
441: 11 8 1
442: 7 5 2
443: 10 6 1
444: 13 12 9
445: 7 6 4
446: 15 7 4
447: 9 6 1
448: 11 6 4
449: 11 6 4
450: 16 12 7
451: 16 10 1
452: 6 5 4
453: 15 6 4
454: 10 9 5
455: 11 6 2
456: 23 11 2
457: 11 8 3
458: 13 10 5
459: 12 5 2
460: 9 5 1
461: 7 6 1
462: 12 11 5
463: 11 8 7
464: 23 9 4
465: 8 3 2
466: 14 11 6
467: 11 6 1
468: 15 9 4
469: 9 5 2
470: 9 8 2
471: 6 3 2
472: 11 3 2
473: 8 6 3
474: 18 11 9
475: 9 8 4
476: 10 8 1
477: 16 15 7
478: 6 4 1
479: 9 7 4
480: 16 13 7
481: 10 9 1
482: 9 6 5
483: 9 6 4
484: 14 2 1
485: 17 16 6
486: 14 8 5
487: 9 4 2
488: 4 3 1
489: 9 6 5
490: 9 7 5
491: 11 6 3
492: 8 7 1
493: 10 5 3
494: 13 5 1
495: 15 9 1
496: 16 5 2
497: 11 9 4
498: 11 9 3
499: 11 6 5
500: 10 6 1
501: 5 4 2
502: 8 5 4
503: 3 2 1
504: 21 14 2
505: 12 8 5
506: 15 12 5
507: 13 6 3
508: 13 8 3
509: 8 7 3
510: 12 10 9
511: 10 8 2
512: 8 5 2
513: 13 10 8
514: 7 5 3
515: 14 7 4
516: 7 5 2
517: 12 10 2
518: 11 3 2
519: 12 8 2
520: 17 13 11
521: 9 7 2
522: 15 13 4
523: 13 6 2
524: 9 5 1
525: 6 4 1
526: 9 5 1
527: 9 7 1
528: 11 6 2
529: 7 4 1
530: 10 7 3
531: 12 6 2
532: 10 4 3
533: 4 3 2
534: 7 5 1
535: 8 6 2
536: 7 5 3
537: 10 2 1
538: 5 2 1
539: 10 5 4
540: 11 6 3
541: 13 10 4
542: 9 3 2
543: 11 7 5
544: 13 9 6
545: 13 8 6
546: 8 2 1
547: 13 7 4
548: 10 5 3
549: 16 4 3
550: 21 17 4
551: 9 4 1
552: 20 5 2
553: 11 4 3
554: 11 8 3
555: 10 9 4
556: 16 10 7
557: 7 6 5
558: 14 9 5
559: 9 7 2
560: 11 9 6
561: 11 7 4
562: 11 4 2
563: 14 7 3
564: 6 3 1
565: 11 6 1
566: 6 5 2
567: 11 9 6
568: 17 11 10
569: 12 10 1
570: 18 12 7
571: 10 5 2
572: 12 8 1
573: 10 6 4
574: 14 9 5
575: 6 5 3
576: 13 4 3
577: 8 3 2
578: 23 22 16
579: 12 9 7
580: 6 4 1
581: 13 7 6
582: 11 6 3
583: 8 6 2
584: 14 13 3
585: 8 3 2
586: 7 5 2
587: 11 6 1
588: 17 16 11
589: 10 4 3
590: 12 3 2
591: 9 6 4
592: 24 19 1
593: 9 8 5
594: 11 10 8
595: 9 2 1
596: 6 5 4
597: 14 12 9
598: 7 6 1
599: 9 8 6
600: 11 10 1
601: 12 4 1
602: 11 8 6
603: 6 4 3
604: 15 6 4
605: 10 7 5
606: 15 7 4
607: 12 9 7
608: 23 6 2
609: 12 9 8
610: 11 10 8
611: 10 4 2
612: 14 10 5
613: 19 10 4
614: 7 2 1
615: 7 6 1
616: 19 10 3
617: 10 9 5
618: 20 13 5
619: 9 8 5
620: 9 2 1
621: 12 6 5
622: 17 12 10
623: 11 10 9
624: 12 9 7
625: 12 8 5
626: 13 5 3
627: 14 10 5
628: 12 9 4
629: 6 5 2
630: 7 4 2
631: 14 8 6
632: 19 13 3
633: 7 2 1
634: 7 5 3
635: 14 10 4
636: 13 8 4
637: 14 9 1
638: 6 5 1
639: 10 4 3
640: 14 3 2
641: 19 5 1
642: 10 9 6
643: 11 3 2
644: 12 11 10
645: 11 8 4
646: 13 12 11
647: 5 4 1
648: 23 22 1
649: 11 5 1
650: 18 15 6
651: 14 13 5
652: 11 9 5
653: 10 8 7
654: 14 11 5
655: 17 15 5
656: 19 18 10
657: 8 7 1
658: 12 10 7
659: 15 4 2
660: 12 4 3
661: 12 11 4
662: 12 6 3
663: 14 11 8
664: 15 4 2
665: 11 6 4
666: 10 7 2
667: 18 7 3
668: 17 12 10
669: 5 4 2
670: 6 5 1
671: 9 6 2
672: 11 6 5
673: 10 9 7
674: 14 9 3
675: 6 3 1
676: 12 5 1
677: 8 4 3
678: 15 5 3
679: 18 12 3
680: 35 30 1
681: 11 9 3
682: 7 3 1
683: 11 6 1
684: 18 13 3
685: 4 3 1
686: 13 12 2
687: 14 12 5
688: 19 14 6
689: 8 6 3
690: 10 7 3
691: 13 6 2
692: 14 6 5
693: 15 8 2
694: 17 13 3
695: 9 4 1
696: 23 10 2
697: 16 12 8
698: 10 9 8
699: 15 10 1
700: 6 5 2
701: 16 4 2
702: 7 3 1
703: 12 7 1
704: 12 5 3
705: 8 6 5
706: 14 11 9
707: 15 8 5
708: 5 4 2
709: 4 3 1
710: 15 14 1
711: 11 8 7
712: 5 4 3
713: 17 10 7
714: 13 7 5
715: 7 4 1
716: 12 11 10
717: 16 7 1
718: 5 2 1
719: 12 9 8
720: 11 8 2
721: 9 6 3
722: 15 4 1
723: 16 13 6
724: 13 8 5
725: 9 6 5
726: 5 4 1
727: 11 8 6
728: 4 3 2
729: 11 5 3
730: 19 15 4
731: 8 6 2
732: 7 4 3
733: 8 7 2
734: 14 13 10
735: 8 7 2
736: 13 8 6
737: 5 4 1
738: 11 9 8
739: 18 16 8
740: 24 12 3
741: 9 8 3
742: 12 4 1
743: 13 12 1
744: 13 11 1
745: 8 7 5
746: 18 13 8
747: 10 6 4
748: 15 5 4
749: 7 6 1
750: 16 9 4
751: 11 3 1
752: 21 20 3
753: 13 8 5
754: 19 14 12
755: 12 10 1
756: 16 9 1
757: 7 6 1
758: 17 12 1
759: 9 3 2
760: 26 13 3
761: 3 2 1
762: 17 7 1
763: 16 14 9
764: 6 5 3
765: 11 10 5
766: 22 19 9
767: 8 7 4
768: 19 17 4
769: 9 7 6
770: 14 5 2
771: 17 15 6
772: 8 6 5
773: 10 8 6
774: 16 14 7
775: 7 6 4
776: 17 12 3
777: 16 10 1
778: 19 16 3
779: 10 8 3
780: 16 8 5
781: 17 16 2
782: 9 3 2
783: 10 7 1
784: 13 9 6
785: 10 9 5
786: 15 6 4
787: 7 6 3
788: 17 10 3
789: 5 2 1
790: 9 7 3
791: 6 4 1
792: 23 17 13
793: 9 4 1
794: 13 12 5
795: 20 15 14
796: 9 4 1
797: 12 10 4
798: 7 6 3
799: 9 6 4
800: 14 9 6
801: 9 7 1
802: 15 13 12
803: 14 9 2
804: 10 9 7
805: 8 7 2
806: 10 9 5
807: 7 6 3
808: 22 3 2
809: 8 3 1
810: 13 12 7
811: 12 10 8
812: 17 6 2
813: 10 3 1
814: 10 4 1
815: 12 10 7
816: 23 15 5
817: 13 8 5
818: 15 7 1
819: 16 9 7
820: 12 7 3
821: 15 11 2
822: 22 16 5
823: 9 6 3
824: 14 11 3
825: 13 7 3
826: 13 8 6
827: 12 10 7
828: 14 13 10
829: 4 3 1
830: 17 10 7
831: 11 8 6
832: 13 5 2
833: 8 3 2
834: 12 7 4
835: 14 7 5
836: 10 9 2
837: 8 6 5
838: 12 8 1
839: 14 6 2
840: 11 5 1
841: 6 5 4
842: 12 8 7
843: 11 10 7
844: 18 16 11
845: 10 8 7
846: 13 12 10
847: 17 13 3
848: 11 4 1
849: 11 9 3
850: 16 7 6
851: 13 10 5
852: 8 5 4
853: 10 7 1
854: 7 5 3
855: 17 3 2
856: 19 10 3
857: 6 5 3
858: 16 10 7
859: 17 15 4
860: 14 11 2
861: 9 5 2
862: 7 6 1
863: 6 3 2
864: 21 10 6
865: 6 4 1
866: 14 13 6
867: 9 5 2
868: 17 12 5
869: 11 7 6
870: 17 16 11
871: 12 4 1
872: 16 7 2
873: 7 3 1
874: 12 7 4
875: 12 8 1
876: 14 8 5
877: 6 5 4
878: 20 9 7
879: 9 5 3
880: 15 7 5
881: 11 8 5
882: 11 5 1
883: 17 16 12
884: 15 11 8
885: 8 7 1
886: 13 9 8
887: 7 6 4
888: 19 18 10
889: 10 8 6
890: 18 13 5
891: 12 10 3
892: 14 7 6
893: 11 8 6
894: 19 7 1
895: 12 8 4
896: 23 21 16
897: 16 8 7
898: 12 5 2
899: 18 15 5
900: 10 7 5
901: 13 7 6
902: 20 16 9
903: 8 7 1
904: 15 11 1
905: 15 9 2
906: 13 11 2
907: 12 10 2
908: 8 7 4
909: 14 4 1
910: 15 9 7
911: 8 6 1
912: 20 5 2
913: 18 12 9
914: 4 2 1
915: 8 6 3
916: 17 12 8
917: 12 10 7
918: 13 12 10
919: 9 7 5
920: 21 7 4
921: 9 8 6
922: 7 6 5
923: 16 14 13
924: 13 10 8
925: 16 15 7
926: 13 6 5
927: 9 7 3
928: 17 11 3
929: 11 4 3
930: 18 14 11
931: 10 9 4
932: 9 2 1
933: 16 6 1
934: 22 6 5
935: 12 11 7
936: 16 13 7
937: 9 5 4
938: 15 6 3
939: 7 5 4
940: 17 16 3
941: 11 6 1
942: 19 14 13
943: 17 14 4
944: 14 3 2
945: 21 4 1
946: 23 11 10
947: 9 6 5
948: 18 16 15
949: 8 3 2
950: 16 14 9
951: 7 6 1
952: 16 9 7
953: 10 7 2
954: 11 8 6
955: 7 6 3
956: 6 5 4
957: 10 9 6
958: 14 9 5
959: 13 10 8
960: 13 9 6
961: 14 9 8
962: 15 8 5
963: 20 9 6
964: 6 4 1
965: 15 4 2
966: 12 9 7
967: 20 8 3
968: 19 18 13
969: 11 10 4
970: 12 5 2
971: 6 2 1
972: 20 13 5
973: 13 6 4
974: 21 10 1
975: 13 4 1
976: 21 4 2
977: 6 3 2
978: 11 7 5
979: 18 16 14
980: 8 7 6
981: 12 6 5
982: 7 6 2
983: 10 4 3
984: 41 23 10
985: 8 4 2
986: 19 12 10
987: 16 13 12
988: 17 3 1
989: 10 4 2
990: 17 16 11
991: 10 9 3
992: 27 14 7
993: 10 8 1
994: 19 9 6
995: 16 11 1
996: 11 8 1
997: 12 6 3
998: 19 15 14
999: 14 11 5
1000: 16 3 2
1001: 5 3 1
1002: 5 3 2
1003: 13 8 3
1004: 17 14 12
1005: 18 16 7
1006: 5 4 3
1007: 8 3 1
1008: 27 13 10
1009: 11 4 2
1010: 10 7 2
1011: 10 7 4
1012: 9 2 1
1013: 9 8 6
1014: 17 11 1
1015: 12 3 1
1016: 15 6 3
1017: 9 4 1
1018: 12 10 5
1019: 10 8 1
1020: 15 12 7
1021: 5 2 1
1022: 14 12 7
1023: 14 8 3
1024: 23 22 9
1025: 9 5 1
1026: 19 14 4
1027: 13 12 6
1028: 9 5 1
1029: 9 8 3
1030: 17 5 1
1031: 12 3 2
1032: 25 18 10
1033: 12 5 2
1034: 13 10 6
1035: 12 6 5
1036: 13 9 6
1037: 12 7 2
1038: 15 7 3
1039: 5 4 2
1040: 26 19 1
1041: 8 5 1
1042: 16 3 1
1043: 10 7 6
1044: 20 8 1
1045: 13 9 6
1046: 16 14 13
1047: 10 8 2
1048: 24 21 3
1049: 11 5 3
1050: 17 9 2
1051: 13 12 10
1052: 8 7 1
1053: 10 9 1
1054: 14 5 4
1055: 15 6 5
1056: 14 9 6
1057: 14 9 4
1058: 14 13 7
1059: 6 3 1
1060: 17 13 8
1061: 10 3 1
1062: 5 2 1
1063: 11 6 2
1064: 18 9 6
1065: 8 7 5
1066: 10 9 3
1067: 13 9 8
1068: 15 8 3
1069: 18 16 8
1070: 15 14 11
1071: 10 7 5
1072: 19 9 8
1073: 12 6 3
1074: 17 14 10
1075: 15 14 5
1076: 17 4 1
1077: 10 9 7
1078: 17 14 4
1079: 17 14 5
1080: 29 12 3
1081: 16 14 2
1082: 14 11 6
1083: 16 7 2
1084: 17 9 5
1085: 10 4 2
1086: 10 9 4
1087: 8 7 3
1088: 22 21 10
1089: 16 8 1
1090: 8 2 1
1091: 12 10 5
1092: 21 20 15
1093: 7 6 1
1094: 5 4 3
1095: 5 4 2
1096: 24 15 6
1097: 13 11 3
1098: 18 15 7
1099: 16 9 1
1100: 19 16 14
1101: 9 7 4
1102: 15 10 8
1103: 13 6 4
1104: 21 12 7
1105: 16 9 6
1106: 16 11 3
1107: 23 11 10
1108: 13 9 6
1109: 17 14 3
1110: 16 6 1
1111: 7 6 4
1112: 15 8 6
1113: 12 9 5
1114: 19 10 6
1115: 18 15 3
1116: 5 4 2
1117: 12 10 4
1118: 9 7 5
1119: 11 5 2
1120: 13 11 5
1121: 12 9 5
1122: 16 13 1
1123: 14 7 3
1124: 22 12 5
1125: 15 8 3
1126: 21 17 13
1127: 9 3 2
1128: 7 3 1
1129: 8 6 5
1130: 11 10 3
1131: 10 6 1
1132: 20 2 1
1133: 11 6 4
1134: 16 13 4
1135: 9 4 2
1136: 9 4 2
1137: 17 12 5
1138: 7 4 3
1139: 13 12 5
1140: 17 16 4
1141: 12 7 3
1142: 12 9 6
1143: 7 2 1
1144: 15 10 1
1145: 16 12 8
1146: 16 9 3
1147: 7 6 3
1148: 25 5 4
1149: 20 17 3
1150: 13 4 1
1151: 15 13 10
1152: 17 11 10
1153: 14 12 2
1154: 9 2 1
1155: 13 6 1
1156: 17 13 11
1157: 8 7 3
1158: 8 4 1
1159: 14 10 5
1160: 30 15 6
1161: 13 11 2
1162: 18 16 11
1163: 11 10 1
1164: 17 14 2
1165: 8 6 1
1166: 17 12 7
1167: 17 13 5
1168: 12 7 2
1169: 13 8 4
1170: 12 11 9
1171: 6 5 1
1172: 15 13 5
1173: 17 14 5
1174: 10 9 2
1175: 12 11 4
1176: 19 2 1
1177: 11 3 1
1178: 11 10 1
1179: 15 8 6
1180: 6 4 3
1181: 17 11 6
1182: 15 11 3
1183: 11 10 5
1184: 5 3 2
1185: 12 3 2
1186: 10 6 5
1187: 13 8 4
1188: 25 16 5
1189: 8 3 1
1190: 20 10 1
1191: 7 6 1
1192: 9 8 7
1193: 10 9 2
1194: 15 14 12
1195: 13 6 5
1196: 17 15 11
1197: 9 8 2
1198: 10 9 5
1199: 16 7 2
1200: 23 8 5
1201: 12 7 5
1202: 13 4 2
1203: 8 4 2
1204: 17 6 2
1205: 4 2 1
1206: 22 13 12
1207: UNFACTORED
1208: 11 10 6
1209: 12 4 3
1210: 9 7 5
1211: 14 7 1
1212: 15 11 2
1213: UNFACTORED
1214: 9 5 4
1215: 13 7 4
1216: 29 12 11
1217: UNFACTORED
1218: 17 14 10
1219: 12 10 6
1220: 18 6 5
1221: 20 13 7
1222: 18 16 1
1223: 8 6 1
1224: 12 9 7
1225: 17 11 4
1226: 20 17 9
1227: 16 13 10
1228: 9 4 1
1229: UNFACTORED
1230: 15 3 1
1231: UNFACTORED
1232: 25 10 2
1233: 11 6 3
1234: 14 11 6
1235: 13 9 8
1236: 11 7 4
1237: UNFACTORED
1238: 12 10 1
1239: 9 3 1
1240: 23 13 6
1241: 13 6 5
1242: 15 13 10
1243: UNFACTORED
1244: 22 19 9
1245: 13 6 1
1246: 11 10 5
1247: 15 13 5
1248: 16 13 3
1249: UNFACTORED
1250: 15 13 10
1251: 13 10 5
1252: 9 5 1
1253: UNFACTORED
1254: 20 16 5
1255: UNFACTORED
1256: 31 30 2
1257: 14 11 7
1258: 9 6 4
1259: UNFACTORED
1260: 15 8 3
1261: 7 4 1
1262: 7 4 2
1263: 12 11 2
1264: 17 11 6
1265: UNFACTORED
1266: 13 10 7
1267: 9 5 2
1268: 12 2 1
1269: 17 10 8
1270: 16 15 5
1271: 13 10 6
1272: 25 9 3
1273: 16 8 6
1274: 15 13 4
1275: 11 10 1
1276: 16 11 2
1277: UNFACTORED
1278: 22 14 13
1279: 16 8 3
1280: 17 15 7
1281: 12 11 5
1282: 22 17 6
1283: UNFACTORED
1284: 15 13 3
1285: 10 9 1
1286: 18 8 5
1287: 7 6 1
1288: 33 24 7
1289: 16 8 1
1290: 10 4 3
1291: UNFACTORED
1292: 15 14 1
1293: 14 9 6
1294: 9 8 2
1295: 10 5 1
1296: 21 15 14
1297: UNFACTORED
1298: 20 19 13
1299: 14 11 5
1300: 17 11 7
1301: 11 10 1
1302: 13 6 1
1303: 16 12 8
1304: 20 17 15
1305: 22 20 2
1306: 11 8 7
1307: 14 12 9
1308: 16 11 1
1309: 8 3 2
1310: 13 11 3
1311: 14 12 9
1312: 25 21 2
1313: 18 8 6
1314: 18 11 6
1315: UNFACTORED
1316: 19 12 11
1317: 17 8 7
1318: 13 12 7
1319: UNFACTORED
1320: 25 11 8
1321: 13 12 1
1322: 15 13 8
1323: 17 16 2
1324: 12 5 1
1325: UNFACTORED
1326: 12 9 2
1327: 10 6 2
1328: 26 17 5
1329: 11 5 2
1330: 11 6 1
1331: UNFACTORED
1332: 16 13 8
1333: 11 3 2
1334: 20 8 5
1335: 7 4 1
1336: 8 3 2
1337: 12 10 1
1338: 19 17 4
1339: UNFACTORED
1340: 16 10 9
1341: 6 5 2
1342: 15 9 7
1343: UNFACTORED
1344: 15 6 1
1345: UNFACTORED
1346: 6 3 2
1347: 10 9 7
1348: 16 12 11
1349: UNFACTORED
1350: 17 15 14
1351: UNFACTORED
1352: 29 27 17
1353: 10 5 1
1354: 7 4 3
1355: 8 4 1
1356: 8 6 3
1357: UNFACTORED
1358: 20 13 4
1359: 8 7 2
1360: 23 21 1
1361: 11 6 2
1362: 24 11 6
1363: 9 5 2
1364: 8 7 3
1365: 11 8 4
1366: 27 22 21
1367: UNFACTORED
1368: 15 9 5
1369: 9 5 4
1370: 10 5 3
1371: 10 9 4
1372: 14 13 5
1373: 15 11 2
1374: 19 5 2
1375: 10 7 1
1376: 23 14 11
1377: 12 10 4
1378: 7 6 3
1379: UNFACTORED
1380: 14 13 11
1381: UNFACTORED
1382: 10 9 2
1383: 16 5 4
1384: 11 5 1
1385: UNFACTORED
1386: 12 8 3
1387: UNFACTORED
1388: 11 8 5
1389: 19 9 4
1390: 19 17 11
1391: 3 2 1
1392: 21 17 6
1393: 12 5 2
1394: 17 13 9
1395: 14 6 5
1396: 29 24 8
1397: 13 8 3
1398: 21 12 11
1399: 18 10 2
1400: 13 10 6
1401: 22 12 3
1402: 15 12 6
1403: UNFACTORED
1404: 19 7 3
1405: 15 10 8
1406: 23 20 2
1407: 9 7 4
1408: 15 12 10
1409: 18 12 5
1410: 18 8 3
1411: 25 14 11
1412: 12 7 4
1413: 20 19 16
1414: 13 3 1
1415: UNFACTORED
1416: 10 9 6
1417: UNFACTORED
1418: 5 3 2
1419: UNFACTORED
1420: 12 6 3
1421: UNFACTORED
1422: 13 7 4
1423: UNFACTORED
1424: 11 6 2
1425: 19 9 4
1426: 11 5 3
1427: 18 17 8
1428: 15 12 8
1429: UNFACTORED
1430: 13 9 1
1431: 8 3 2
1432: 11 10 6
1433: UNFACTORED
1434: 11 8 6
1435: 15 10 9
1436: 21 17 13
1437: 7 6 4
1438: 24 23 22
1439: UNFACTORED
1440: 14 13 7
1441: 5 3 1
1442: 9 4 2
1443: UNFACTORED
1444: 16 13 10
1445: 13 10 3
1446: 17 3 2
1447: UNFACTORED
1448: 25 22 6
1449: 21 9 7
1450: 16 12 3
1451: UNFACTORED
1452: 17 14 5
1453: UNFACTORED
1454: 11 6 2
1455: 17 9 8
1456: 9 4 2
1457: 9 7 1
1458: 21 19 13
1459: 16 14 9
1460: 18 17 10
1461: 12 5 3
1462: 13 5 2
1463: UNFACTORED
1464: 23 14 1
1465: UNFACTORED
1466: 11 10 3
1467: UNFACTORED
1468: 16 11 8
1469: 20 18 14
1470: 8 5 1
1471: 7 3 1
1472: 11 4 1
1473: 14 11 5
1474: 17 11 10
1475: UNFACTORED
1476: 11 8 3
1477: UNFACTORED
1478: 25 12 5
1479: 17 8 7
1480: 17 4 3
1481: UNFACTORED
1482: 15 12 1
1483: UNFACTORED
1484: 15 11 7
1485: 23 16 11
1486: 13 10 9
1487: 14 8 2
1488: 21 15 14
1489: 14 3 1
1490: 15 13 8
1491: 9 5 2
1492: 14 13 8
1493: UNFACTORED
1494: 13 12 9
1495: 21 11 7
1496: 13 11 4
1497: UNFACTORED
1498: 17 11 10
1499: UNFACTORED
1500: 14 2 1

(5..1206 time = 1h, 2min, 25,858 ms.
1207..1499 time = 1h, 18min, 51,597 ms.
1500 time = 20,539 ms)