[njzibpmo] Factorization of a prime gap

Let p = 1425172824437699411 discovered by Tomás Oliveira e Silva. p starts a largest known maximal prime gap, of length 1476. p - 1 = 2 * 5 * 217363 * 655664866807.

Here are the least prime factors of every composite in the prime gap. Of course, "2" occurs every other entry.
? for(i=p+1, nextprime(p+1)-1, print1(" ", factorint(i)[1,1]))

2 17 2 3 2 79 2 269 2 3 2 37019 2 5 2 3 2 27065459 2 769 2 3 2 5 2 2273 2 3 2 13 2 3187 2 3 2 11 2 7 2 3 2 661 2 5 2 3 2 107 2 73 2 3 2 5 2 13 2 3 2 181 2 4993 2 3 2 7 2 1567 2 3 2 993253 2 5 2 3 2 787 2 7 2 3 2 5 2 53 2 3 2 311 2 29 2 3 2 9168407 2 47 2 3 2 11 2 5 2 3 2 7 2 2776183 2 3 2 5 2 139 2 3 2 211 2 7 2 3 2 83 2 101 2 3 2 1433 2 5 2 3 2 17 2 122250157 2 3 2 5 2 11 2 3 2 7 2 8673107 2 3 2 449 2 67 2 3 2 5527 2 5 2 3 2 11 2 523 2 3 2 5 2 37 2 3 2 71 2 116447 2 3 2 13 2 19 2 3 2 7 2 5 2 3 2 113383 2 2251 2 3 2 5 2 7 2 3 2 531359 2 11 2 3 2 43 2 251 2 3 2 169243 2 5 2 3 2 281 2 97 2 3 2 5 2 7130407 2 3 2 17 2 114889 2 3 2 24023 2 7 2 3 2 1999631 2 5 2 3 2 109 2 23 2 3 2 5 2 29 2 3 2 31 2 657147307 2 3 2 7 2 11 2 3 2 437279 2 5 2 3 2 3347471 2 7 2 3 2 5 2 476975971 2 3 2 11 2 19 2 3 2 23 2 17 2 3 2 16747 2 5 2 3 2 7 2 68957909 2 3 2 5 2 89391721 2 3 2 101 2 7 2 3 2 547 2 258112223 2 3 2 13 2 5 2 3 2 508547617 2 59 2 3 2 5 2 31437613 2 3 2 7 2 61 2 3 2 11 2 13 2 3 2 647 2 5 2 3 2 19 2 47 2 3 2 5 2 28591 2 3 2 229551227 2 29399 2 3 2 47843 2 23 2 3 2 7 2 5 2 3 2 10238791 2 11 2 3 2 5 2 7 2 3 2 13 2 181 2 3 2 67 2 239 2 3 2 11 2 5 2 3 2 5431 2 29 2 3 2 5 2 13 2 3 2 9533 2 605609 2 3 2 31 2 7 2 3 2 428873 2 5 2 3 2 59 2 151 2 3 2 5 2 11 2 3 2 79 2 18127 2 3 2 7 2 73 2 3 2 19 2 5 2 3 2 11 2 7 2 3 2 5 2 47933 2 3 2 53 2 17 2 3 2 3623 2 31 2 3 2 41 2 5 2 3 2 7 2 19 2 3 2 5 2 23 2 3 2 54078419 2 7 2 3 2 17 2 3805721 2 3 2 2777 2 5 2 3 2 3347 2 43 2 3 2 5 2 11895823 2 3 2 7 2 6477203 2 3 2 13 2 555829 2 3 2 23 2 5 2 3 2 193 2 283 2 3 2 5 2 796871 2 3 2 383 2 13 2 3 2 19 2 11 2 3 2 7 2 5 2 3 2 97 2 37 2 3 2 5 2 7 2 3 2 11 2 89 2 3 2 9869179 2 79 2 3 2 31 2 5 2 3 2 17 2 179 2 3 2 5 2 197 2 3 2 149 2 47 2 3 2 241 2 7 2 3 2 29 2 5 2 3 2 503 2 13 2 3 2 5 2 41 2 3 2 107 2 81199 2 3 2 7 2 431 2 3 2 307 2 5 2 3 2 3701 2 7 2 3 2 5 2 17 2 3 2 19 2 53 2 3 2 577 2 61 2 3 2 13 2 5 2 3 2 7 2 11 2 3 2 5 2 122448839 2 3 2 17 2 7 2 3 2 47 2 13 2 3 2 11 2 5 2 3 2 37 2 182641 2 3 2 5 2 2797 2 3 2 7 2 443 2 3 2 8093 2 29 2 3 2 14083 2 5 2 3 2 1051 2 2903569 2 3 2 5 2 11 2 3 2 13 2 23 2 3 2 70373 2 17 2 3 2 7 2 5 2 3 2 11 2 199 2 3 2 5 2 7 2 3 2 1693 2 37 2 3 2 29 2 3272879 2 3 2 17 2 5 2 3 2 23 2 641 2 3 2 5 2 337 2 3 2 263 2 11 2 3 2 475681 2 7 2 3 2 2475089 2 5 2 3 2 13 2 31 2 3 2 5 2 64811 2 3 2 107201 2 919 2 3 2 7 2 1571 2 3 2 109 2 5 2 3 2 463 2 7 2 3 2 5 2 73 2 3 2 56209 2 41 2 3 2 101 2 11 2 3 2 32569 2 5 2 3 2 7 2 23 2 3 2 5 2 83 2 3 2 11 2 7 2 3 2 13 2 510707 2 3 2 61 2 5 2 3 2 173099 2 113 2 3 2 5 2 19 2 3 2 7 2 13 2 3 2 23 2 293 2 3 2 2679401 2 5 2 3 2 137 2 41183 2 3 2 5 2 173 2 3 2 29 2 17 2 3 2 11 2 739 2 3 2 7 2 5 2 3 2 47 2 53 2 3 2 5 2 7 2 3 2 197 2 2711 2 3 2 17 2 59 2 3 2 19 2 5 2 3 2 257 2 11 2 3 2 5 2 31 2 3 2 977 2 197927 2 3 2 40459499 2 7 2 3 2 11 2 5 2 3 2 756919 2 19 2 3 2 5 2 331 2 3 2 844999 2 79 2 3 2 7 2 107 2 3 2 13 2 5 2 3 2 157 2 7 2 3 2 5 2 11 2 3 2 118043 2 7069 2 3 2 53 2 13 2 3 2 103 2 5 2 3 2 7 2 677 2 3 2 5 2 89 2 3 2 2371801 2 7 2 3 2 19 2 41 2 3 2 3329 2 5 2 3 2 233 2 271 2 3 2 5 2 90067 2 3 2 7 2 11 2 3 2 113 2 2687 2 3 2 37 2 5 2 3 2 73 2 373 2 3 2 5 2 13 2 3 2 67 2 38851 2 3 2 2027 2 457 2 3 2 7 2 5 2 3 2 43 2 2819 2 3 2 5 2 7 2 3 2 17 2 31 2 3 2 66533 2 11 2 3 2 23 2 5 2 3 2 13 2 5087 2 3 2 5 2 37 2 3 2 11 2 3313 2 3 2 163 2 7 2 3 2 6263 2 5 2 3 2 4003 2 29 2 3 2 5 2 71 2 3 2 47 2 1808887 2 3 2 7 2 17 2 3 2 107 2 5 2 3 2 61 2 7 2 3 2 5 2 131 2 3 2 165553 2 281 2 3 2 11 2 53 2 3 2 17 2 5 2 3 2 7 2 7121 2 3 2 5 2 151 2 3 2 15541 2 7 2 3 2 31 2 563 2 3 2 223 2 5 2 3 2 244639 2 11 2 3 2 5 2 409 2 3 2 7 2 59 2 3 2 349279013 2 167 2 3 2 11 2 5 2 3 2 79 2 17 2 3 2 5 2 3907 2 3 2 149211191 2 19 2 3 2 865102109 2 31 2 3 2 7 2 5 2 3 2 71 2 13 2 3 2 5 2 7 2 3 2 1675759 2 30689 2 3 2

Any number with a minimal factor larger than 1477 could be considered "lucky". There are 104 such numbers.

The largest value, or the most "RSA" on this list is p + 1446 = 865102109 * 1647404173.

Here are the second smallest prime factors, not counting prime powers (e.g., higher powers of 2). Fortunately, none of the composites are a perfect power of a prime. We give "0" if the second smallest happens to be the largest, in order to save space.
? for(i=p+1, nextprime(p+1)-1, s=factorint(i); if(matsize(s)[1]==2, print1(" 0"), print1(" ", s[2,1])))

3 0 7 5 29 359 3 0 5 7 28343261 599681 3 11 733 42697 7 0 3 29567 1401017 31 233 7 3 3889 101 37 5 23 3 0 53 5 135497 17 3 71 5 109 7219 552239 3 43 7 0 11 191 3 76597 47 7 17 223 3 47501 61 11 5 2113 3 137273 29 5 37 23311 3 0 5 17 41 0 3 19 3803 23 14229931 530339 3 11 263 13 541 31 3 154003181 7 307 5 0 3 0 19 5 13 0 3 89 5 1307 7 37 3 17 113227 271 0 13 3 0 479 19 11 59 3 15191663 4007 61 5 0 3 23 73 5 4219 0 3 31487641 5 43 19 0 3 13 11 7 0 277847 3 0 21881 19037 7 14489 3 31 13 0 5 19 3 0 41 5 17 0 3 37208207 5 13 6961 2294141 3 7 901963 81421 131 23 3 809 7 17 13 263183 3 1801 31 7 5 569 3 0 0 5 7 137 3 5077 5 11 23 47 3 41 617 73 0 0 3 0 11 193 18400981 13331 3 17 19 29 5 0 3 13 7 5 18236357 729571 3 59334833 5 7 521 0 3 10039 13 19 7 456979 3 8731 367 59 199 7 3 0 23 13 5 61 3 0 79 5 11 0 3 0 5 37 13 0 3 0 7 11 17 0 3 0 30334589 7 3181 13 3 0 11 12583 5 1187 3 0 133949 5 227 41 3 3610367 5 8727539 24337 0 3 389369 95222429 47 37 0 3 13 17 67 50129 93787 3 0 7 53 5 0 3 43 13 5 277 4423 3 138395689 5 6803323 7 0 3 90071 863 13 41 163 3 0 19 11 137 29 3 0 97 103 5 509 3 31 11 5 1597 4103797 3 0 5 19 1381 17 3 11 43 7 73094669 0 3 0 53 23 7 4937 3 0 80963 41 5 107069 3 149 31 5 89 491 3 0 5 1783 42190319 0 3 7 23 17 11 27241 3 4969 7 29 3464183 18906557 3 223547 96181549 7 5 0 3 0 17 5 7 0 3 37 5 25111 79 823 3 53 0 15383 13 0 3 17 29 293 0 127 3 19 13757 8419 5 0 3 1721 7 5 31 821 3 102587 5 7 43 199 3 397 19 0 7 0 3 41 557 0 11 7 3 1879 0 229 5 0 3 0 496163 5 463 4525091 3 83 5 137 17 0 3 71 7 1319 23 1231 3 0 179 7 19 43 3 107 439 17 5 1630843 3 42953 677 5 13 3756229 3 5783 5 23 314299 76607 3 761 17 1063 173 13 3 157 5014951 227 46573181 0 3 0 7 7307 5 0 3 155013841 23 5 0 15259 3 0 5 11 7 419 3 13 1979 3923 29 113 3 9086881 11 101 139 103 3 0 13 44790377 5 0 3 11 89 5 61 37 3 0 5 13 1913 306421 3 257 4201 7 0 120691 3 67 73 51613 7 11 3 0 0 19 5 0 3 0 4229 5 11 673 3 0 5 17 0 277 3 7 29 11 19 409 3 479 7 4009849 719 709 3 0 11 7 5 2411 3 601 43 5 7 61 3 0 5 0 31 551207 3 17 13 53 1049 73300373 3 5059 151 0 15271 83 3 155608049 67 13 5 0 3 0 7 5 0 0 3 0 5 7 11 24749 3 19 41 43 7 271 3 0 23 11 1523 7 3 0 37 21701 5 76829 3 2381 11 5 17 5711 3 127 5 173 0 139 3 11 7 389 71 15619 3 0 33851 7 4723223 2715737 3 0 10458841 35797 5 1223 3 0 13 5 303755029 11 3 0 5 44027 19 401 3 31 293 13 11 0 3 0 8447 311 2179 8854667 3 89245879 7 11 5 23 3 233 223 5 379 2083 3 2069 5 29 7 43 3 101 31 0 140177 95428999 3 103 1237 521 23 2683 3 0 1453 71 5 167 3 662047 211 5 10733 2529913 3 19 5 199193 67 1117 3 143802179 5189 7 17 41 3 2700583 13 151 7 272383 3 4154621 19 0 5 73 3 0 0 5 0 339653 3 5623 5 83 0 1499123 3 7 11 19 13 1951 3 0 7 4806853 47 0 3 186783721 163 7 5 89 3 97 10008533 5 7 118399 3 43 5 27783311 1511 59 3 967 563 7728067 0 31 3 29401 397 864289 0 19 3 13 6829 1907 5 4691 3 0 7 5 25219 0 3 0 5 7 1229 0 3 437297821 0 529519 7 0 3 1409 11 13 2801 7 3 467 113 96017 5 643 3 19 83 5 13 0 3 1423 5 499 23 0 3 8831 7 17 263752847 19936793 3 71 19 7 0 11 3 740533 47 217421 5 0 3 8233 17 5 11 229 3 0 5 19 61 140333 3 13 3229 11 167 1543 3 17 31 127 151 790879 3 15583433 7 0 5 1321951 3 7620227 29 5 7297 0 3 73061 5 13 7 0 3 47 3203 103 43 19 3 139 3853 31 13 17 3 227 887 149 5 1153 3 29 0 5 181 2897 3 0 5 163 11 17167 3 193 0 7 18539 0 3 0 2269 11 7 128237 3 19183 53 17 5 37 3 0 11 5 607 947 3 557 5 72883 0 0 3 7 13 97 21523 179 3 0 7 1151 0 31 3 0 667867 7 5 99059297 3 1579 311 5 7 478421 3 45751 5 859 13 71 3 10957 0 0 11 353 3 40475353 0 23 19 13 3 66343 971 11 5 1546241 3 0 7 5 41 0 3 1367 5 7 751307 214518319 3 37 23 0 7 275923 3 13 281 73 17 7 3 43 89 29 5 21467 3 0 13 5 47 0 3 23 5 17 0 1920379 3 61 7 13 33623969 0 3 0 37 7 11 379 3 38832191 17 0 5 0 3 0 2803 5 201281 0 3 48193 5 60919 0 83 3 17 11 0 38121449 2969 3 109 1747 47 163 23 3 29 7 19 5 0 3 12853 10672573 5 73 181 3 241 5 5051 7 287849 3 13309 61 283 19 11 3 5591 13 43 67 6269 3 839 193 27197 5 0 3 0 2371 5 17 59 3 863 5 11 503 0 3 0 0 7 13 877 3 2161 11 17 7 29 3 0 0 14939 5 13 3 12301 23 5 43 0 3 0 5 0 100641559 461 3 7 131 61 2207 6373 3 25981 7 251 29 11 3 17 41609 7 5 0 3 0 19 5 7 3877 3 0 5 5639 0 0 3 12281 71 11 197 108193 3 7810801 167 13 347 28751 3 149 11 1709 5 41 3 0 7 5 13 337543 3 0 5 7 19 0 3 0 97231 1373 7 0 3 0 29 137 2267 7 3 0 749744609 83 5 19 3 0 31 5 23 0 3 1093 5 0 11 0 3 13 7 4463 0 0 3 0 17 7 4447 155657 3 307 13 23 5 8517409 3 0 11 5 257 14629 3 19 5 13 3511 367 3 11 0 1091 4711453 0 3 101 23 41 13 89 3 0 7 109 5 1480627 3 75553 59 5 31 13 3 103 5 37 7 173 3 23 27739 19 11 29 3 1997773 83 82837 0 459262901 3 0 43 11 5 17159 3 13 41 5 19 211 3 0 5 0 353 17837 3 97 13 7 29 0 3 0 31513 0 7 19 3 44279 1213 13 5 23 3 32423 17 5 827 0 3 619 5 43 13 7175461 3 7 0 29 199 5273 3 127 7 113 11 13 3 12649709 73 7 5 0 3 157 1621 5 7 0 3 421 5 593 7927 12821 3 83 11 23 109 883 3 61 19 4783 43 17 3 11 4561 0 5 0 3 1441529 7 5 0

The largest entry on this list, not very interesting because of the artificial zeros, is p + 1287 = 2 * 749744609 * 950438861. Largest even RSA?

Here are the penultimate factors
? for(i=p+1, nextprime(p+1)-1, s=factorint(i); t=matsize(s); print1(" ", s[t[1]-1,1]))

601 17 10603141 47 1400023 1100161 11009827 269 193 4621 28343261 599681 3 2281 47279 42697 63199 27065459 313 29567 1401017 7193 27527 14831 121067 12289 7639 277 53569 13967 5351 3187 19447 2028109 135497 317 397 439 1015871 1096553 239357 552239 83 43 69661 3 13136791 21737 9677 76597 38651 67680593 31 82633 38453 723617 61 3433 218887 25801 21379 137273 110248231 405683 15859 23311 61151 1567 8147 887 26437043 993253 1493 7561 588433 971 14229931 530339 1243093 8741 263 941 541 31 5689 154003181 43 307 5 311 67 29 359069 72863267 168212117 9168407 36373 1811 64381 1307 373 37 977 17 113227 3259 2 664331 78487 2776183 1429 41 2339 59 6971 15191663 84499 61 207479 211 432743 103 2517401 11 253969 83 3 31487641 7 1327 19 1433 8537 11257 252313 7 2 277847 6883 122250157 21881 19037 7 270709 90001 180503 13 3 1607 2609 3024227 8673107 163243 2579 5939 449 1399 37208207 5 127 64693 2294141 14969 79 901963 135241 131 151 19 5297 799979 6473 10259 263183 2293111 41651 16360229 17264243 409 619 57982723 116447 2 241 587 937 15199931 9811 1619 4271 152267 153529 3 1213 617 73 2 113383 13 2251 6329 67579 18400981 23813 613253 17 30825649 31 112247 531359 1656131 8179 2447 23 18236357 729571 74857 59334833 857 20707 27799 169243 265709 10039 571 1995013 44381 456979 101 8731 184999 59 2176309 11 604589 7130407 29 487 15299 6890713 6461027 114889 1420093 5 47051 24023 3 7 10883 37 5669 1999631 6790831 5 4709021 832063 1097 109 43 23 30334589 8599 37589 6709 84389 29 431 12583 69341 1187 1573549 657147307 133949 55903 6857 4273 4259 3610367 1194421 8727539 1600061 437279 23 465679 95222429 13687 3559 3347471 4261 461 1598581 30911 50129 1335277 323027 476975971 257 423061 158077 11 401 113 34313 7 7013 21221 283 138395689 451279 6803323 10463 16747 34123 90071 182389 8291 41 163 2357 68957909 38377 858373 1277 9029 691037 89391721 839 743 23 12143 211 3931 47 2341 280451 4103797 1264213 258112223 10453 631 1059557 73 11696771 3607 662339 379 73094669 508547617 5 59 27008027 4193221 317 4937 13 31437613 80963 691 137831 107069 45833 6324883 31 227743 359 491 109 13 107 4174861 42190319 647 349 6320227 1851217 17 6228413 27241 63977 4969 499 1545097 3464183 18906557 3 223547 96181549 1361 143401 229551227 10286117 29399 71 1087 5279 47843 5519 240943 13697 79757 29761 1368811 3 18313 2 15383 9341 10238791 59 2903 1299377 375257 2 77527 6791 2072563 13757 3153317 52354789 13 9439 1721 7681 7528993 241 821 207481 102587 131 191 103 199 94999 397 114281 3 7 5431 14731 50599 557 3 11 71537 3 6037 2 696517 60945091 9533 128287 605609 496163 89 463 4525091 109001 83 42649 137 17 428873 142591 71 455419 2302453 126589891 1231 28097 151 16857391 37 19 373 114601 358427 106759 17 7 1630843 163 42953 3253147 545549 6011 3756229 8699 5783 23333 23 314299 169933 10691 761 101377 1063 173 688133 5 17789 5014951 1847 46573181 5 3 47933 4211 7307 281 53 26959 155013841 23 7 2 15259 6193963 31 51407 11 97 419 632713 36767 1979 3923 1637 1609 223259 9086881 180413 5653 1059221 41011 4597807 23 37182511 44790377 5 54078419 127 211 267433 5 19403 16823 1033 3805721 31 773 7784459 306421 11 4965089 4597 10831 2 120691 347 60271 73 149749 53 11 3 11895823 2 47 615799 7 7358903 6477203 13859 1566827 1901 408211 55343 555829 26125903 14461 2 277 2503 74601187 109 59 1254031 17417 13 479 1591621 4009849 719 709 7351 796871 137 367 61949 2411 22721 166403 164363 4889497 11783 421 263 11 2087 3 24671 551207 24169 29 991 104717 1049 73300373 953 96907 9397 3 15271 1839809 1113491 155608049 48449 36433 5 11 7901 89 1920917 220877 2 9869179 103 79 190367 45127 113 24749 29 163 313 43 2290889 1831 52561 179 84047 1109 3574309 13 23251 197 659 21701 5 76829 3533573 2381 19 683 2767 5711 239 7603 53 15607 2 43201 3691 23 88469 2420339 12347 15619 5 13 3051869 12401 4723223 2715737 40169 41 10458841 95089 14653 1223 111577 81199 1873 109 303755029 911 23 431 2299571 44027 53279 108553 661 59 16253 13 83 3701 42491 7 8447 311 2179 8854667 97 89245879 79 28949 6311 251 2063 349 498973 359 54829 2083 41 2621 907 29 37 929 17 18061 928849 3 140177 95428999 125929 73823 1237 103993 486589 2683 13 122448839 1453 253381 569 167 1745789 662047 2039491 598189 10733 2529913 3 19 196387 199193 13829 28657 67139 143802179 5189 31 17 41 5 2700583 134070539 587 29917 305551 331 4154621 19 3 14438531 73 457 443 2 881 2 339653 7 583229 769 375527 2 1499123 5197 7 181711 571 31247 22901 5488607 2903569 13183309 4806853 109 5 54673 186783721 163 52571 1114447 1760131 257 3733 10008533 157 269 118399 746117 2029 30119 27783311 1511 131 66697 967 2939 7728067 2 21313 179 29401 397 864289 2 19 2393 29023 13001 14087 6247 4691 1856747 37 173 5 25219 29 373 3272879 13 61 1229 17 2237 437297821 2 529519 4679 23 455443 247769 2999 13 321383 7039 787 36749 113 96017 17 643 137 103703 283 5 32077 475681 3 629029 48647 853 27337 2475089 24767 8831 5347 40849 263752847 19936793 1613 71 10450507 7 2 11 521357 740533 39372679 217421 89 107201 374837 8233 17 107 388879 347 241 1571 65719 7203839 8242693 140333 7 963223 3229 439 167 1543 4973 17 5101 130079 151 790879 3 15583433 140681 3 19 1321951 39869 7620227 5805391 2111 365159 101 7283 73061 5 854897 419 32569 96493 152459 6121 223 30013 8685683 1559 13297 4703 343913 1061323 49297 233 1627 2147767 2437 5 78707 10631 1283 2 8689 2887 330131 411779 510707 7 163 1447 17167 93923 24208787 2 5449 147607 173099 809 113 2269 474547 383 128237 2467243 19183 317 23279 36433559 152213 47 13 1933931 5 1997 947 28411 9629 99317 72883 2 2679401 3767 11 19 3061 730589 1657 307 41183 107 1151 2 41 13557679 173 667867 4759 688217 99059297 200407 1579 37337 532267 7 478421 437681 45751 199 74489 379531 2707 10837 10957 2 3 5581181 33773 3643 40475353 2 5843 8873749 20411 31 112909 3797 5531 11243 1546241 1500703 2711 8581 449 829 17 13 1367 22853 54547 751307 214518319 5477 10601 564407 3 7 275923 479 11611801 743 22717 12041 151 18858971 857 5895931 29 2791 21467 19 197927 37307 269 1103 40459499 54941 4220789 2215903 17 2 1920379 113 205967 325181 260761 33623969 756919 5 19 37 2409137 401 232741 4416007 38832191 126827 3 13 844999 3292873 79 2803 103583 201281 7 879863 48193 43 78283 2 83 937 10517167 3207779 3 38121449 166693 431 34649 2053 2741 23909 65179 1065059 1451 18869 31081 51929 118043 12186319 12853 10672573 7 13457 4099 11 5483 5 32839 116000501 287849 3 13309 3677 283 15199 613 5 5591 2987911 6359 10663577 6269 2399 6491 193 27197 12163 2371801 12109 7 13597 1857461 46639 8377 3 863 1819843 1254257 5390291 3329 11821 5 2 52069 1801357 877 173 2161 31091 655927 88423 191 19 90067 2 14939 1104427 322669 227693 176753 601 5 6977 113 7 2687 825337 3 100641559 4021 653 8693 131 523 55807 6373 41 25981 5610463 1976383 84523 23011627 823 47 41609 1787 35027 67 1987 38851 246889 4127 81119 370813 40483 457 2309 6199 2 7 17 3262187 10607 11 821 108193 673 7810801 167 7159 4057 28751 761 12854623 13940959 1709 11027 41 3 31 3183989 102317 13 337543 9929 11 19319 3617 92821 23 47 5 1365583 7135907 174761 13 18341383 5087 472391 1171 511289 292183 3751343 37 749744609 137209 5 33113173 293 3313 2221 4362961 165233 163 3541 1093 1804493 3 4729 6263 13763 3431867 10369 4463 2 4003 1133131 29 7817 828371 269561 155657 3 264359 13 11047 429281 8517409 47857 1808887 773 31 74821 14629 181 19 1789 13 13649711 693037 10211 1249 2 1091 4711453 61 92357 101 23 41 441547 9295849 10651 131 2843 509 5821 1480627 187177 75553 2584949 1097 3856141 26703199 357437 103 587 419 7 437743 1355281 661 27739 383 571 1861 143413 1997773 248797 82837 2 459262901 2960387 151 87993263 12487 499 17159 3527 13 41 239 191 211 305267 563 139681 3 13615163 17837 3 9520127 18523 164251 16447 244639 61 11 31513 3 7 5623571 16567 44279 1307 1426097 5 226169 53 32423 38873 16993 827 349279013 31 619 702349 349 233 7175461 340777 155473 2 29 9041 5273 4324711 547 536509 36683 89 55799 23159 12649709 5119 6949 24247 149211191 101 359 1621 197 246899 865102109 194003 64763 5 33343 7927 51229 17189 2851 29 8237 3343 3391 994489 20323 1499 4783 6277 50821 14159 72689 4561 3 267581 1675759 389161 1441529 2352079 8069 2

Extremely lopsided factorizations (the opposite of "RSA"), where the composite is exactly twice a prime number happen at p + i for i = 107 183 447 515 567 635 671 783 791 827 855 963 975 1035 1091 1307 1335 1475. Note that p + 1475 is the last member of the prime gap.
? for(i=p+1, nextprime(p+1)-1, s=factorint(i); t=matsize(s); if((s[1,1]==2) &&(s[1,2]==1) &&(t[1]==2), print1(" ", i-p)))

The smoothest (minimizing the largest factor) is p + 1228 = 3 * 7 * 139 * 199 * 397 * 1471 * 1787 * 2351.

The largest prime factor appearing with an exponent larger than 1 is 113, at p + 754 = 3 * 5 * 113^2 * 598189 * 12438871.
? z=0; for(i=p+1, nextprime(p+1)-1, s=factorint(i); t=matsize(s); for(j=1,t[1], if(s[j,2]>1,l=s[j,1]; if(l>z,z=l;loc=i-p)))); loc

Graphs would probably improve this post. Cut and paste the numbers?