Factorizations of prime(n)! / prime(n)# + 1 for n = 1 to 26 =========================================================== Contents: page Table 2 n = 1,2,3 p = 2,3,5 3 n = 4,5,6 p = 7,11,13 4 n = 7,8 p = 17,19 5 n = 9,10 p = 23,29 6 n = 11 p = 31 7 n = 12 p = 37 8 n = 13 p = 41 9 n = 14 p = 43 10 n = 15 p = 47 11 n = 16 p = 53 12 n = 17 p = 59 13 n = 18 p = 61 14 n = 19 p = 67 15 n = 20 p = 71 16 n = 21 p = 73 17 n = 22 p = 79 18 n = 23 p = 83 19 n = 24 p = 89 20 n = 25 p = 97 21 n = 26 p = 101 22 n = 27 p = 103 23 All factorizations by using Dario Alpern's ECM-program http://www.alpertron.com.ar/ECM.HTM A103890.txt Reinhard Zumkeller (reinhard.zumkeller@lhsystems.com), Feb 20 2005 page 2 +----+------------+------------+------------+------------+------------+ | | A103890(n) | A103891(n) | A103892(n) | A103893(n) | A103894(n) | | n +------------+------------+------------+------------+------------+ | | | prime factors | number of | | | p=prime(n) | smallest | greatest | # distinct | divisors | | | |------------+------------+------------+------------+ | | x=p!/p#+1 | A020639(x) | A006530(x) | A001221(x) | A000005(x) | +----+------------+------------+------------+------------+------------+ | 1 | 2 | 2 | 2 | 1 | 2 | | 2 | 2 | 2 | 2 | 1 | 2 | | 3 | 5 | 5 | 5 | 1 | 2 | | 4 | 25 | 5 | 5 | 1 | 3 | | 5 | 17281 | 11 | 1571 | 2 | 4 | | 6 | 207361 | 7 | 2693 | 3 | 8 | | 7 | 696729601 | 19 | 15427 | 3 | 8 | | 8 |12541132801 | 31 | 27749 | 4 | 16 | | 9 | [Len=15] | 27893 | 4154449757 | 2 | 4 | | 10 | [Len=22] | 223 | [Len=15] | 3 | 8 | | 11 | [Len=23] | 554797 | [Len=17] | 2 | 4 | | 12 | [Len=31] | 10333 | [Len=14] | 4 | 16 | | 13 | [Len=36] | 29 | [Len=19] | 4 | 16 | | 14 | [Len=37] | 61 | [Len=19] | 5 | 32 | | 15 | [Len=42] | 28652131 | [Len=23] | 3 | 8 | | 16 | [Len=51] | 293049347 | [Len=42] | 2 | 4 | | 17 | [Len=59] | 7731517 | [Len=41] | 3 | 8 | | 18 | [Len=61] | 1483 | [Len=41] | 4 | 16 | | 19 | [Len=70] | 73 | [Len=39] | 6 | 64 | | 20 | [Len=76] | [Len=12] | [Len=22] | 5 | 32 | | 21 | [Len=78] | 67 | [Len=63] | 5 | 32 | | 22 | [Len=87] | 440939 | [Len=33] | 5 | 32 | | 23 | [Len=93] | 1321 | [Len=60] | 5 | 32 | | 24 | [Len=102] | 94099 | [Len=36] | 6 | 64 | | 25 | [Len=116] | 89 | [Len=99] | 5 | 32 | | 26 | [Len=122] | 107 | [Len=104] | 4 | 16 | | 27 | [Len=124] | 281 | [Len=102] | 5 | 32 | +----+------------+------------+------------+------------+------------+ Len = number of digits in decimal representation (A055642), e.g. A103890(9) = 23!/23#+1 = 115880067072001 with Len=15. ----------- ------------------- page 3 2! / 2# + 1 n = 1: prime(1) = 2 ----------- ------------------- 2! = 2; 2# = 2 2! / 2# + 1 = 2 omega = 1; Omega = 1; mu = -1; tau = 2 sigma = 3; phi = 1 ======================================================================= ----------- ------------------- 3! / 3# + 1 n = 2: prime(2) = 3 ----------- ------------------- 3! = 6; 3# = 6 3! / 3# + 1 = 2 omega = 1; Omega = 1; mu = -1; tau = 2 sigma = 3; phi = 1 ======================================================================= ----------- ------------------- 5! / 5# + 1 n = 3: prime(3) = 5 ----------- ------------------- 5! = 120; 5# = 30 5! / 5# + 1 = 5 omega = 1; Omega = 1; mu = -1; tau = 2 sigma = 6; phi = 4 ======================================================================= ----------- ------------------- page 4 7! / 7# + 1 n = 4: prime(4) = 7 ----------- ------------------- 7! = 5040; 7# = 210 7! / 7# + 1 = 25 Factorization #1: 5^2 omega = 1; Omega = 2; mu = 0; tau = 3 sigma = 31; phi = 20 ======================================================================= ------------- -------------------- 11! / 11# + 1 n = 5: prime(5) = 11 ------------- -------------------- 11! = 39 916800; 11# = 2310 11! / 11# + 1 = 17281 Factorization #1: 11 #2: 1571 omega = 2; Omega = 2; mu = +1; tau = 4 sigma = 18864; phi = 15700 ======================================================================= ------------- -------------------- 13! / 13# + 1 n = 6: prime(6) = 13 ------------- -------------------- 13! = 6227 020800; 13# = 30030 13! / 13# + 1 = 207361 Factorization #1: 7 #2: 11 #3: 2693 omega = 3; Omega = 3; mu = -1; tau = 8; sigma = 258624; phi = 161520 ======================================================================= ------------- -------------------- page 5 17! / 17# + 1 n = 7: prime(7) = 17 ------------- -------------------- 17! = 355 687428 096000; 17# = 510510 [Len=15] 17! / 17# + 1 = 696 729601 Factorization #1: 19 #2: 2377 #2: 15427 omega = 3; Omega = 3; mu = -1; tau = 8 sigma = 733755680; phi = 659739168 ======================================================================= ------------- -------------------- 19! / 19# + 1 n = 8: prime(8) = 19 ------------- -------------------- 19! = 121645 100408 832000; 19# = 9 699690 [Len=18] 19! / 19# + 1 = 12541 132801 [Len=11] Factorization #1: 31 #2: 61 #3: 239 #4: 27749 omega = 4; Omega = 4; mu = +1; tau = 16 sigma = 13213 440000; phi = 11887 243200 [Len=11] [Len=11] ======================================================================= ------------- -------------------- page 6 23! / 23# + 1 n = 9: prime(9) = 23 ------------- -------------------- 23! = 25852 016738 884976 640000; 23# = 223 092870 [Len=23] 23! / 23# + 1 = 115 880067 072001 [Len=15] Factorization #1: 27893 #2: 4154 449757 [Len=10] omega = 2; Omega = 2; mu = +1; tau = 4 sigma = 115 884221 549652; phi = 115 875912 594352 [Len=15] [Len=15] ======================================================================= ------------- ---------------------- 29! / 29# + 1 n = 10: prime(10) = 29 ------------- ---------------------- 29! = 8 841761 993739 701954 543616 000000 [Len=31] 29# = 6469 693230 [Len=10] 29! / 29# + 1 = 1366 643159 020339 200001 [Len=22] Factorization #1: 223 #2: 40429 #3: 151 585363 459003 [Len=15] omega = 3; Omega = 3; mu = -1; tau = 8 sigma = 1372 805558 801047 105280 [Len=22] phi = 1360 481062 410358 294032 [Len=22] ======================================================================= ------------- ---------------------- page 7 31! / 31# + 1 n = 11: prime(11) = 31 ------------- ---------------------- 31! = 8222 838654 177922 817725 562880 000000 [Len=34] 31# = 200560 490130 [Len=12] 31! / 31# + 1 = 40999 294770 610176 000001 [Len=23] Factorization #1: 554797 #2: 73899 633146 196133 [Len=17] omega = 2; Omega = 2; mu = +1; tau = 4 sigma = 40999 368670 243322 750932 [Len=23] phi = 40999 220870 977029 249072 [Len=23] ======================================================================= ------------- ---------------------- page 8 37! / 37# + 1 n = 12: prime(12) = 37 ------------- ---------------------- 37! = 13 763753 091226 345046 315979 581580 902400 000000 [Len=44] 37# = 7 420738 134810 [Len=13] 37! / 37# + 1 = 1 854768 736099 424576 471040 000001 [Len=31] Factorization #1: 10333 #2: 17551 #3: 412 876421 #4: 24 770877 398207 [Len=14] omega = 4; Omega = 4; mu = +1; tau = 16 sigma = 1 855053 929165 456908 669711 085568 [Len=31] phi = 1 854483 563489 396099 991763 032000 [Len=31] ======================================================================= ------------- ---------------------- page 9 41! / 41# + 1 n = 13: prime(13) = 41 ------------- ---------------------- 41! = 33 452526 613163 807108 170062 053440 751665 152000 000000 [Len=50] 41# = 304 250263 527210 [Len=15] 41! / 41# + 1 = 109950 690675 973888 893203 251200 000001 [Len=36] Factorization #1: 29 #2: 743 #3: 3 570470 221379 #4: 1 429176 066859 677377 [Len=19] omega = 4; Omega = 4; mu = +1; tau = 16 sigma = 113895 178720 398409 713058 206857 404800 [Len=36] phi = 106016 408292 727234 808287 504322 403328 [Len=36] ======================================================================= ------------- ---------------------- page 10 43! / 43# + 1 n = 14: prime(14) = 43 ------------- ---------------------- 43! = 60415 263063 373835 637355 132068 513997 507264 512000 000000 [Len=53] 43# = 13082 761331 670030 [Len=17] 43! / 43# + 1 = 4 617929 008390 903333 514536 550400 000001 [Len=37] Factorization #1: 61 #2: 613 #3: 1 100653 #4: 14 481017 #5: 7 748318 379505 746557 [Len=19] omega = 5; Omega = 5; mu = -1; tau = 32 sigma = 4 701294 182073 054820 056713 442085 236768 [Len=37] phi = 4 534810 991759 912394 035394 805790 730240 [Len=37] ======================================================================= ------------- ---------------------- page 11 47! / 47# + 1 n = 15: prime(15) = 47 ------------- ---------------------- 47! = 258623 241511 168180 642964 355153 611979 969197 632389 120000 000000 [Len=60] 47# = 614889 782588 491410 [Len=18] 47! / 47# + 1 = 420600 974084 243475 616503 989010 432000 000001 [Len=42] Factorization #1: 28 652131 #2: 677643 511121 #3: 21662 673895 172922 098651 [Len=23] omega = 3; Omega = 3; mu = -1; tau = 8 sigma = 420600 988764 434556 002649 278043 833454 083808 [Len=42] phi = 420600 959404 052395 273684 047768 731734 440000 [Len=42] ======================================================================= ------------- ---------------------- page 12 53! / 53# + 1 n = 16: prime(16) = 53 ------------- ---------------------- 53! = 4274 883284 060025 564298 013753 389399 649690 343788 366813 724672 000000 000000 [Len=70] 53# = 32 589158 477190 044730 [Len=20] 53! / 53# + 1 = 131 175012 912718 250806 592304 873426 282086 400000 000001 [Len=51] Factorization #1: 293 049347 #2: 447620 901583 917369 406703 727865 417431 168683 [Len=42] omega = 2; Omega = 2; mu = +1; tau = 4 sigma = 131 175013 360339 152390 509674 280130 009951 817724 218032 [Len=51] phi = 131 175012 465097 349222 674935 466722 554220 982275 781972 [Len=51] ======================================================================= ------------- ---------------------- page 13 59! / 59# + 1 n = 17: prime(17) = 59 ------------- ---------------------- 59! = 138 683118 545689 835737 939019 720389 406345 902876 772687 432540 821294 940160 000000 000000 [Len=81] 59# = 1922 760350 154212 639070 [Len=22] 59! / 59# + 1 = 72127 095056 108748 061547 940676 488545 036232 818688 000000 000001 [Len=59] Factorization #1: 7 731517 #2: 347785 835587 #3: 26823 894708 473887 265117 477378 744683 922119 [Len=41] omega = 3; Omega = 3; mu = -1; tau = 8 sigma = 72127 104385 286772 370615 263357 328839 468994 235929 957013 958080 [Len=59] phi = 72127 085726 930723 806128 407412 596025 133706 356204 227941 020368 [Len=59] ======================================================================= ------------- ---------------------- page 14 61! / 61# + 1 n = 18: prime(18) = 61 ------------- ---------------------- 61! = 507580 213877 224798 800856 812176 625227 226004 528988 036003 099405 939480 985600 000000 000000 [Len=84] 61# = 117288 381359 406970 983270 [Len=24] 61! / 61# + 1 = 4 327625 703366 524883 692876 440589 312702 173969 121280 000000 000001 [Len=61] Factorization #1: 1483 #2: 7127 #3: 7 008809 876989 #4: 58419 455175 821973 834615 220346 798867 768249 [Len=41] omega = 4; Omega = 4; mu = +1; tau = 16 sigma = 4 331151 484674 765440 661947 496309 232509 572841 941130 209291 760000 [Len=61] phi = 4 324100 740959 994221 941239 110029 140624 838195 439409 501080 221568 [Len=61] ======================================================================= ------------- ---------------------- page 15 67! / 67# + 1 n = 19: prime(19) = 67 ------------- ---------------------- 67! = 36471 110918 188685 288249 859096 605464 427167 635314 049524 593701 628500 267962 436943 872000 000000 000000 [Len=95] 67# = 7 858321 551080 267055 879090 [Len=25] 67! / 67# + 1 = 4641 081518 632318 859492 593303 493155 822657 704661 332315 340800 000000 000001 [Len=70] Factorization #1: 73 #2: 463 #3: 292069 #4: 1 485559 #5: 1684 513433 610479 #6: 187 873487 957710 251112 719111 051932 885411 [Len=39] omega = 6; Omega = 6; mu = +1; tau = 64 sigma = 4714 838542 729049 076050 633579 797208 711316 188355 531786 143035 007987 712000 [Len=70] phi = 4567 599725 971282 082891 714564 566159 491163 232096 471032 203915 836492 551680 [Len=70] ======================================================================= ------------- ---------------------- page 16 71! / 71# + 1 n = 20: prime(20) = 71 ------------- ---------------------- 71! = 850478 588567 862317 521167 644239 926010 288584 608120 796235 886430 763388 588680 378079 017697 280000 000000 000000 [Len=102] 71# = 557 940830 126698 960967 415390 [Len=27] 71! / 71# + 1 = 1524 316813 979598 806211 747344 599292 098393 696518 967985 650532 352000 000000 000001 [Len=76] Factorization #1: 337809 273983 #2: 5 530881 877187 #3: 31 734470 040281 #4: 10638 349187 975557 #5: 2416 594580 545944 177593 [Len=22] omega = 5; Omega = 5; mu = -1; tau = 32 sigma = 1524 316813 984434 943298 474956 333546 576264 648335 538343 359387 327016 533991 436288 [Len=76] phi = 1524 316813 974762 669125 021667 221733 459675 731931 826330 862642 208638 346306 309120 [Len=76] ======================================================================= ------------- ---------------------- page 17 73! / 73# + 1 n = 21: prime(21) = 73 ------------- ---------------------- 73! = 4470 115461 512684 340891 257138 125051 110076 800700 282905 015819 080092 370422 104067 183317 016903 680000 000000 000000 [Len=106] 73# = 40729 680599 249024 150621 323470 [Len=29] 73! / 73# + 1 = 109750 810606 531114 047245 808811 149031 084346 149365 694966 838329 344000 000000 000001 [Len=78] Factorization #1: 67 #2: 1861 #3: 8539 #4: 678253 #5: 151 980484 198948 823551 696833 505973 745524 167842 228692 010030 901969 [Len=63] omega = 5; Omega = 5; mu = -1; tau = 32 sigma = 111461 952791 070229 304005 829509 981995 239183 374051 698309 575695 450501 005140 643200 [Len=78] phi = 108041 831366 305927 381253 035388 442403 103883 841243 974396 796749 181454 414328 007680 [Len=78] ======================================================================= ------------- ---------------------- page 18 79! / 79# + 1 n = 22: prime(22) = 79 ------------- ---------------------- 79! = 894 618213 078297 528685 144171 539831 652069 808216 779571 907213 868063 227837 990693 501860 533361 810841 010176 000000 000000 000000 [Len=117] 79# = 3 217644 767340 672907 899084 554130 [Len=31] 79! / 79# + 1 = 278 035108 834491 952374 200462 632195 223836 131498 049515 522896 518347 698995 200000 000000 000001 [Len=87] Factorization #1: 440939 #2: 229 918267 #3: 1 800121 214093 #4: 3745 828933 595186 733991 167749 #5: 406 722276 544872 969063 670718 259161 [Len=33] omega = 5; Omega = 5; mu = -1; tau = 32 sigma = 278 035740 596245 831054 283943 139751 395511 341437 099068 804021 560315 995511 808224 120644 240000 [Len=87] phi = 278 034477 072743 559408 826422 312502 996081 319735 887546 173796 087019 613617 593641 525557 076480 [Len=87] ======================================================================= ------------- ---------------------- page 19 83! / 83# + 1 n = 23: prime(23) = 83 ------------- ---------------------- 83! = 39455 239697 206586 511897 471180 120610 571436 503407 643446 275224 357528 369751 562996 629334 879591 940103 770870 906880 000000 000000 000000 [Len=125] 83# = 267 064515 689275 851355 624017 992790 [Len=33] 83! / 83# + 1 = 147 736735 430295 643813 555157 824243 254137 566832 803590 568246 293989 233338 089472 000000 000000 000001 [Len=93] Factorization #1: 1321 #2: 3011 #3: 19433 #4: 11078 093307 398182 946017 #5: 172532 158443 057699 772164 532525 607875 550686 109683 184687 080411 [Len=60] omega = 5; Omega = 5; mu = -1; tau = 32 sigma = 147 905285 933934 490801 739062 831597 514902 659513 292302 182595 675127 417705 095702 712339 039900 934016 [Len=93] phi = 147 568275 772045 730870 660656 103068 115714 931370 817962 736601 560391 309055 520335 862966 744914 944000 [Len=93] ======================================================================= ------------- ---------------------- page 20 89! / 89# + 1 n = 24: prime(24) = 89 ------------- ---------------------- 89! = 16507 955160 908461 081216 919262 453619 309839 666236 496541 854913 520707 833171 034378 509739 399912 570787 600662 729080 382999 756800 000000 000000 000000 [Len=137] 89# = 23768 741896 345550 770650 537601 358310 [Len=35] 89! / 89# + 1 = 694523 725020 825051 446420 532149 209597 411955 055040 671457 338706 588955 510775 037065 953280 000000 000000 000001 [Len=102] Factorization #1: 94099 #2: 6 218647 #3: 18 494666 981437 #4: 58 443131 174093 #5: 1489 840216 743290 665508 300119 #6: 737032 243754 315571 280907 352212 738723 [Len=36] omega = 6; Omega = 6; mu = +1; tau = 64 sigma = 694531 217483 012975 969966 712428 735690 840774 686507 093707 428631 852834 372563 356026 843310 828454 438466 048000 [Len=102] phi = 694516 232561 010884 497880 528005 969849 492270 547142 607000 819360 921525 637135 665128 270808 367104 457490 345216 [Len=102] ======================================================================= ------------- ---------------------- page 21 97! / 97# + 1 n = 25: prime(25) = 97 ------------- ---------------------- 97! = 96 192759 682482 119853 328425 949563 698712 343813 919172 976158 104477 319333 745612 481875 498805 879175 589072 651261 284189 679678 167647 067832 320000 000000 000000 000000 [Len=152] 97# = 2 305567 963945 518424 753102 147331 756070 [Len=37] 97! / 97# + 1 = 41 721936 280667 021214 842775 197740 704647 938666 856859 336935 449248 035545 074425 279071 887258 996965 376000 000000 000000 000001 [Len=116] Factorization #1: 89 #2: 647 #3: 6449 #4: 196 829639 #5: 570 804377 775579 113845 140084 714923 228047 667959 084711 538825 242983 631969 831787 242913 416490 381542 591177 [Len=99] omega = 5; Omega = 5; mu = -1; tau = 32 sigma = 42 262484 390795 535964 385608 824561 285729 225004 300548 209910 025087 617481 701713 435358 941355 345731 682014 205638 584130 880000 [Len=116] phi = 41 183002 663048 748699 366087 065051 230963 707944 195859 745484 380638 809409 292395 123488 020829 230389 455182 392152 590320 996352 [Len=116] ======================================================================= --------------- ----------------------- page 22 101! / 101# + 1 n = 26: prime(26) = 101 --------------- ----------------------- 101! = 9425 947759 838359 420851 623124 482936 749562 312794 702543 768327 889353 416977 599316 221476 503087 861591 808346 911623 490003 549599 583369 706302 603264 000000 000000 000000 000000 [Len=160] 101# = 232 862364 358497 360900 063316 880507 363070 [Len=39] 101! / 101# + 1 = 40 478622 579503 143982 640460 496848 031649 430094 584524 928694 772860 444085 831207 405755 545018 678855 807795 200000 000000 000000 000001 [Len=122] Factorization #1: 107 #2: 55849 #3: 149254 503023 #4: 45 383616 852383 776893 383271 999181 668387 799180 429227 611638 947076 007267 619009 534006 636655 641206 385330 905509 [Len=104] omega = 4; Omega = 4; mu = +1; tau = 16 sigma = 40 857659 024288 777402 572245 450856 182519 700033 868239 753275 000458 574292 961434 075097 260541 866282 046576 406950 010437 487179 232000 [Len=122] phi = 40 099599 682140 946911 614205 695341 811686 879283 545595 342878 575455 874945 119753 377762 020285 405143 362661 496054 172339 331808 060288 [Len=122] ======================================================================= --------------- ----------------------- page 23 103! / 103# + 1 n = 27: prime(27) = 103 --------------- ----------------------- 103! = 99 029007 164861 804075 467152 545817 733490 901658 221144 924830 052805 546998 766658 416222 832141 441073 883538 492653 516385 977292 093222 882134 415149 891584 000000 000000 000000 000000 [Len=164] 103# = 23984 823528 925228 172706 521638 692258 396210 [Len=41] 103! / 103# + 1 = 4128 819503 109320 686229 326970 678499 228241 869647 621542 726866 831765 296754 783155 387065 591905 243292 395110 400000 000000 000000 000001 [Len=124] Factorization #1: 281 #2: 3319 #3: 1 668791 #4: 4105 856767 #5: 646110 176498 902974 386727 572279 803881 142700 113371 182675 202263 115519 317905 465027 783917 805971 035221 955047 [Len=102] omega = 5; Omega = 5; mu = -1; tau = 32 sigma = 4144 763717 708205 414161 133929 596567 483324 785760 226779 966769 110657 760575 287772 515967 298881 367447 072017 580301 305707 130219 397120 [Len=124] phi = 4112 884161 675768 311450 162462 605776 551124 239143 923991 470610 810667 232589 292582 391771 784953 632685 037986 452701 569579 908113 737600 [Len=124] =======================================================================