we crudely attempt to extend OEIS A057713, the smallest prime factor of P# - 1, where P# is the primorial function. previously, primorial+1, which explains more details.
some factors were found with Pari/GP and GMP-ECM. other factors were gathered from https://oeis.org/A057713/b057713.txt , https://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/matha103.htm , and http://factordb.com . the factor for nthprime(217)#-1 = primorial(1327) - 1 was not previously known.
the next prime is a factor for indices 2, 4, and no known further indices.
the prime factors listed below are not guaranteed to be smallest factors, because both Pollard rho and the Elliptic Curve Method is probabilistic. this is why above we described this calculation as "crudely attempt to extend".
we ran GMP-ECM with its autoincrement flag (-I), and for completely unfactored entries below, we report the last B1 we tried.
nthprime(2)# - 1 = 3# - 1 = 5 is prime
nthprime(3)# - 1 = 5# - 1 = 29 is prime
nthprime(4)# - 1 = 7# - 1 = 209 is divisible by 11
nthprime(5)# - 1 = 11# - 1 is prime
nthprime(6)# - 1 = 13# - 1 is prime
nthprime(7)# - 1 = 17# - 1 is divisible by 61
nthprime(8)# - 1 = 19# - 1 is divisible by 53
nthprime(9)# - 1 = 23# - 1 is divisible by 37
nthprime(10)# - 1 = 29# - 1 is divisible by 79
nthprime(11)# - 1 = 31# - 1 is divisible by 228737
nthprime(12)# - 1 = 37# - 1 is divisible by 229
nthprime(13)# - 1 = 41# - 1 is prime
nthprime(14)# - 1 = 43# - 1 is divisible by 141269
nthprime(15)# - 1 = 47# - 1 is divisible by 191
nthprime(16)# - 1 = 53# - 1 is divisible by 87337
nthprime(17)# - 1 = 59# - 1 is divisible by 27600124633
nthprime(18)# - 1 = 61# - 1 is divisible by 1193
nthprime(19)# - 1 = 67# - 1 is divisible by 163
nthprime(20)# - 1 = 71# - 1 is divisible by 260681003321
nthprime(21)# - 1 = 73# - 1 is divisible by 313
nthprime(22)# - 1 = 79# - 1 is divisible by 163
nthprime(23)# - 1 = 83# - 1 is divisible by 139
nthprime(24)# - 1 = 89# - 1 is prime
nthprime(25)# - 1 = 97# - 1 is divisible by 66683
nthprime(26)# - 1 = 101# - 1 is divisible by 2990092035859
nthprime(27)# - 1 = 103# - 1 is divisible by 15649
nthprime(28)# - 1 = 107# - 1 is divisible by 17515703
nthprime(29)# - 1 = 109# - 1 is divisible by 719
nthprime(30)# - 1 = 113# - 1 is divisible by 295201
nthprime(31)# - 1 = 127# - 1 is divisible by 15098753
nthprime(32)# - 1 = 131# - 1 is divisible by 10172884549
nthprime(33)# - 1 = 137# - 1 is divisible by 20962699238647
nthprime(34)# - 1 = 139# - 1 is divisible by 4871
nthprime(35)# - 1 = 149# - 1 is divisible by 673
nthprime(36)# - 1 = 151# - 1 is divisible by 311
nthprime(37)# - 1 = 157# - 1 is divisible by 1409
nthprime(38)# - 1 = 163# - 1 is divisible by 1291
nthprime(39)# - 1 = 167# - 1 is divisible by 331
nthprime(40)# - 1 = 173# - 1 is divisible by 1450184819
nthprime(41)# - 1 = 179# - 1 is divisible by 23497
nthprime(42)# - 1 = 181# - 1 is divisible by 711427
nthprime(43)# - 1 = 191# - 1 is divisible by 521
nthprime(44)# - 1 = 193# - 1 is divisible by 673
nthprime(45)# - 1 = 197# - 1 is divisible by 519577
nthprime(46)# - 1 = 199# - 1 is divisible by 1372062943
nthprime(47)# - 1 = 211# - 1 is divisible by 56543
nthprime(48)# - 1 = 223# - 1 is divisible by 811
nthprime(49)# - 1 = 227# - 1 is divisible by 182309
nthprime(50)# - 1 = 229# - 1 is divisible by 53077
nthprime(51)# - 1 = 233# - 1 is divisible by 641
nthprime(52)# - 1 = 239# - 1 is divisible by 349
nthprime(53)# - 1 = 241# - 1 is divisible by 389
nthprime(54)# - 1 = 251# - 1 is divisible by 565921
nthprime(55)# - 1 = 257# - 1 is divisible by 777041
nthprime(56)# - 1 = 263# - 1 is divisible by 1176221
nthprime(57)# - 1 = 269# - 1 is divisible by 757
nthprime(58)# - 1 = 271# - 1 is divisible by 2341
nthprime(59)# - 1 = 277# - 1 is divisible by 44753
nthprime(60)# - 1 = 281# - 1 is divisible by 303665792864057405074693
nthprime(61)# - 1 = 283# - 1 is divisible by 7027
nthprime(62)# - 1 = 293# - 1 is divisible by 4111
nthprime(63)# - 1 = 307# - 1 is divisible by 1571
nthprime(64)# - 1 = 311# - 1 is divisible by 166661291
nthprime(65)# - 1 = 313# - 1 is divisible by 917337320621
nthprime(66)# - 1 = 317# - 1 is prime
nthprime(67)# - 1 = 331# - 1 is divisible by 7037216702504839
nthprime(68)# - 1 = 337# - 1 is prime
nthprime(69)# - 1 = 347# - 1 is divisible by 1609
nthprime(70)# - 1 = 349# - 1 is divisible by 33409
nthprime(71)# - 1 = 353# - 1 is divisible by 18422659
nthprime(72)# - 1 = 359# - 1 is divisible by 1069
nthprime(73)# - 1 = 367# - 1 is divisible by 1880785297
nthprime(74)# - 1 = 373# - 1 is divisible by 2705777831
nthprime(75)# - 1 = 379# - 1 is divisible by 6361
nthprime(76)# - 1 = 383# - 1 is divisible by 5407
nthprime(77)# - 1 = 389# - 1 is divisible by 104707
nthprime(78)# - 1 = 397# - 1 is divisible by 599
nthprime(79)# - 1 = 401# - 1 is divisible by 4139
nthprime(80)# - 1 = 409# - 1 is divisible by 35447
nthprime(81)# - 1 = 419# - 1 is divisible by 1986323
nthprime(82)# - 1 = 421# - 1 is divisible by 761
nthprime(83)# - 1 = 431# - 1 is divisible by 234277373353
nthprime(84)# - 1 = 433# - 1 is divisible by 1447641166619113176017
nthprime(85)# - 1 = 439# - 1 is divisible by 33073
nthprime(86)# - 1 = 443# - 1 is divisible by 16054891
nthprime(87)# - 1 = 449# - 1 is divisible by 1594387
nthprime(88)# - 1 = 457# - 1 is divisible by 4012079
nthprime(89)# - 1 = 461# - 1 is divisible by 23107891172756940484396791068944577893302431569651
nthprime(90)# - 1 = 463# - 1 is divisible by 631
nthprime(91)# - 1 = 467# - 1 is divisible by 11379319098532079
nthprime(92)# - 1 = 479# - 1 is divisible by 2609
nthprime(93)# - 1 = 487# - 1 is divisible by 494441
nthprime(94)# - 1 = 491# - 1 is divisible by 17929
nthprime(95)# - 1 = 499# - 1 is divisible by 569
nthprime(96)# - 1 = 503# - 1 is divisible by 3300480581931696902302732092200380863444990184404360359
nthprime(97)# - 1 = 509# - 1 is divisible by 593759939
nthprime(98)# - 1 = 521# - 1 is divisible by 3666321961
nthprime(99)# - 1 = 523# - 1 is divisible by 563
nthprime(100)# - 1 = 541# - 1 is divisible by 6014627
nthprime(101)# - 1 = 547# - 1 is divisible by 1777
nthprime(102)# - 1 = 557# - 1 is divisible by 53563779143263213754862638009468737
nthprime(103)# - 1 = 563# - 1 is divisible by 7867
nthprime(104)# - 1 = 569# - 1 is divisible by 12101
nthprime(105)# - 1 = 571# - 1 is divisible by 10753
nthprime(106)# - 1 = 577# - 1 is divisible by 1093
nthprime(107)# - 1 = 587# - 1 is divisible by 2113
nthprime(108)# - 1 = 593# - 1 is divisible by 10859
nthprime(109)# - 1 = 599# - 1 is divisible by 4003709
nthprime(110)# - 1 = 601# - 1 is divisible by 54273959359
nthprime(111)# - 1 = 607# - 1 is divisible by 69473
nthprime(112)# - 1 = 613# - 1 is divisible by 5659
nthprime(113)# - 1 = 617# - 1 is divisible by 5591
nthprime(114)# - 1 = 619# - 1 is divisible by 296071
nthprime(115)# - 1 = 631# - 1 is divisible by 7927
nthprime(116)# - 1 = 641# - 1 is divisible by 881
nthprime(117)# - 1 = 643# - 1 is divisible by 715927
nthprime(118)# - 1 = 647# - 1 is divisible by 109357
nthprime(119)# - 1 = 653# - 1 is divisible by 109673
nthprime(120)# - 1 = 659# - 1 is divisible by 76948279003051
nthprime(121)# - 1 = 661# - 1 is divisible by 224859040921421
nthprime(122)# - 1 = 673# - 1 is divisible by 38200012273
nthprime(123)# - 1 = 677# - 1 is divisible by 2803
nthprime(124)# - 1 = 683# - 1 is divisible by 8259773239673410178219
nthprime(125)# - 1 = 691# - 1 is divisible by 2063
nthprime(126)# - 1 = 701# - 1 is divisible by 2801
nthprime(127)# - 1 = 709# - 1 is divisible by 660924521623121209230107
nthprime(128)# - 1 = 719# - 1 is divisible by 315643
nthprime(129)# - 1 = 727# - 1 is divisible by 8101
nthprime(130)# - 1 = 733# - 1 is divisible by 5239123655163246348151
nthprime(131)# - 1 = 739# - 1 is divisible by 118387
nthprime(132)# - 1 = 743# - 1 is divisible by 57455597751970998877176249003156521
nthprime(133)# - 1 = 751# - 1 is divisible by 808823054396726705208944607100082721043
nthprime(134)# - 1 = 757# - 1 is divisible by 6637
nthprime(135)# - 1 = 761# - 1 is divisible by 7677996300827
nthprime(136)# - 1 = 769# - 1 is divisible by 1160227
nthprime(137)# - 1 = 773# - 1 is divisible by 638233
nthprime(138)# - 1 = 787# - 1 is divisible by 640227319
nthprime(139)# - 1 = 797# - 1 is divisible by 953
nthprime(140)# - 1 = 809# - 1 is divisible by 21991
nthprime(141)# - 1 = 811# - 1 is divisible by 3199720721
nthprime(142)# - 1 = 821# - 1 is divisible by 42844194023
nthprime(143)# - 1 = 823# - 1 is divisible by 1579031
nthprime(144)# - 1 = 827# - 1 is divisible by 14503
nthprime(145)# - 1 = 829# - 1 is divisible by 26987
nthprime(146)# - 1 = 839# - 1 is divisible by 1751693
nthprime(147)# - 1 = 853# - 1 is divisible by 20548103533
nthprime(148)# - 1 = 857# - 1 is divisible by 35573
nthprime(149)# - 1 = 859# - 1 is divisible by 246016739
nthprime(150)# - 1 = 863# - 1 is divisible by 746027341427
nthprime(151)# - 1 = 877# - 1 is divisible by 911
nthprime(152)# - 1 = 881# - 1 is divisible by 41134919254014829
nthprime(153)# - 1 = 883# - 1 is divisible by 1110311750119070976651453707
nthprime(154)# - 1 = 887# - 1 is divisible by 11257
nthprime(155)# - 1 = 907# - 1 is divisible by 983
nthprime(156)# - 1 = 911# - 1 is divisible by 109547
nthprime(157)# - 1 = 919# - 1 is divisible by 1181
nthprime(158)# - 1 = 929# - 1 is divisible by 30376905151
nthprime(159)# - 1 = 937# - 1 is divisible by 37811
nthprime(160)# - 1 = 941# - 1 is divisible by 1433
nthprime(161)# - 1 = 947# - 1 is divisible by 2153
nthprime(162)# - 1 = 953# - 1 is divisible by 38212861741
nthprime(163)# - 1 = 967# - 1 is divisible by 14771753
nthprime(164)# - 1 = 971# - 1 is divisible by 5737
nthprime(165)# - 1 = 977# - 1 is divisible by 14083
nthprime(166)# - 1 = 983# - 1 is divisible by 26633
nthprime(167)# - 1 = 991# - 1 is prime
nthprime(168)# - 1 = 997# - 1 is divisible by 106859
nthprime(169)# - 1 = 1009# - 1 is divisible by 521317
nthprime(170)# - 1 = 1013# - 1 is divisible by 112593828893724701195685754565695150560753586811
nthprime(171)# - 1 = 1019# - 1 is divisible by 41299
nthprime(172)# - 1 = 1021# - 1 is divisible by 389783
nthprime(173)# - 1 = 1031# - 1 is divisible by 28771
nthprime(174)# - 1 = 1033# - 1 is divisible by 11217725881
nthprime(175)# - 1 = 1039# - 1 is divisible by 1296546826589465489576348219482737598352223058436635171
nthprime(176)# - 1 = 1049# - 1 is divisible by 31127898314092269938684484007635727
nthprime(177)# - 1 = 1051# - 1 is divisible by 14969
nthprime(178)# - 1 = 1061# - 1 is divisible by 784423
nthprime(179)# - 1 = 1063# - 1 is divisible by 291503
nthprime(180)# - 1 = 1069# - 1 is divisible by 1327
nthprime(181)# - 1 = 1087# - 1 is divisible by 17299
nthprime(182)# - 1 = 1091# - 1 is unfactored, 458 digits, last B1=6448937
nthprime(183)# - 1 = 1093# - 1 is divisible by 330015654119
nthprime(184)# - 1 = 1097# - 1 is divisible by 256284923
nthprime(185)# - 1 = 1103# - 1 is divisible by 1283
nthprime(186)# - 1 = 1109# - 1 is divisible by 24407
nthprime(187)# - 1 = 1117# - 1 is divisible by 4157
nthprime(188)# - 1 = 1123# - 1 is divisible by 23469739
nthprime(189)# - 1 = 1129# - 1 is divisible by 4943
nthprime(190)# - 1 = 1151# - 1 is divisible by 1555074301957555828730151725339
nthprime(191)# - 1 = 1153# - 1 is divisible by 1424525332337
nthprime(192)# - 1 = 1163# - 1 is divisible by 1151633336245487868340531
nthprime(193)# - 1 = 1171# - 1 is divisible by 2437343
nthprime(194)# - 1 = 1181# - 1 is divisible by 227629
nthprime(195)# - 1 = 1187# - 1 is divisible by 102859
nthprime(196)# - 1 = 1193# - 1 is unfactored, 501 digits, last B1=5997526
nthprime(197)# - 1 = 1201# - 1 is divisible by 9234487357
nthprime(198)# - 1 = 1213# - 1 is divisible by 17347517031847
nthprime(199)# - 1 = 1217# - 1 is divisible by 27814739077
nthprime(200)# - 1 = 1223# - 1 is divisible by 3787687
nthprime(201)# - 1 = 1229# - 1 is unfactored, 516 digits, last B1=5788781
nthprime(202)# - 1 = 1231# - 1 is divisible by 167687809
nthprime(203)# - 1 = 1237# - 1 is divisible by 2311595977
nthprime(204)# - 1 = 1249# - 1 is divisible by 163236519283
nthprime(205)# - 1 = 1259# - 1 is divisible by 73979777
nthprime(206)# - 1 = 1277# - 1 is divisible by 1273457
nthprime(207)# - 1 = 1279# - 1 is divisible by 1383607
nthprime(208)# - 1 = 1283# - 1 is divisible by 19687
nthprime(209)# - 1 = 1289# - 1 is divisible by 1811
nthprime(210)# - 1 = 1291# - 1 is divisible by 4319941
nthprime(211)# - 1 = 1297# - 1 is divisible by 43913
nthprime(212)# - 1 = 1301# - 1 is divisible by 43329742931
nthprime(213)# - 1 = 1303# - 1 is divisible by 179866327
nthprime(214)# - 1 = 1307# - 1 is divisible by 440342341
nthprime(215)# - 1 = 1319# - 1 is divisible by 391579
nthprime(216)# - 1 = 1321# - 1 is divisible by 260189
nthprime(217)# - 1 = 1327# - 1 is divisible by 954765457247896375172877804352140973
nthprime(218)# - 1 = 1361# - 1 is divisible by 5320092072242699
nthprime(219)# - 1 = 1367# - 1 is divisible by 20681
nthprime(220)# - 1 = 1373# - 1 is divisible by 756923
nthprime(221)# - 1 = 1381# - 1 is divisible by 48299
nthprime(222)# - 1 = 1399# - 1 is divisible by 220307
nthprime(223)# - 1 = 1409# - 1 is divisible by 3911
nthprime(224)# - 1 = 1423# - 1 is divisible by 34217
nthprime(225)# - 1 = 1427# - 1 is divisible by 21637487
nthprime(226)# - 1 = 1429# - 1 is divisible by 6421
nthprime(227)# - 1 = 1433# - 1 is divisible by 86263
nthprime(228)# - 1 = 1439# - 1 is divisible by 3557
nthprime(229)# - 1 = 1447# - 1 is unfactored, 604 digits, last B1=11689233
nthprime(230)# - 1 = 1451# - 1 is divisible by 341816863
nthprime(231)# - 1 = 1453# - 1 is divisible by 8317
nthprime(232)# - 1 = 1459# - 1 is divisible by 7691
nthprime(233)# - 1 = 1471# - 1 is divisible by 1770787
nthprime(234)# - 1 = 1481# - 1 is divisible by 5331773
nthprime(235)# - 1 = 1483# - 1 is divisible by 1631647
No comments :
Post a Comment