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 All factorizations by using Dario Alpern's ECM-program http://www.alpertron.com.ar/ECM.HTM A103855.txt Reinhard Zumkeller (reinhard.zumkeller@lhsystems.com), Feb 18 2005 page 2 +----+------------+------------+------------+------------+------------+ | | A103855(n) | A103856(n) | A103857(n) | A103858(n) | A103859(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 | 1 | 1 | 1 | 0 | 1 | | 2 | 1 | 1 | 1 | 0 | 1 | | 3 | 91 | 7 | 13 | 2 | 4 | | 4 | 4831 | 4831 | 4831 | 1 | 2 | | 5 | 39914491 | 3673 | 10867 | 2 | 4 | | 6 | 6226990771 | 16349 | 380879 | 2 | 4 | | 7 | [Len=15] | 5240507 | 67872713 | 2 | 4 | | 8 | [Len=18] | 4159 | [Len=14] | 2 | 4 | | 9 | [Len=23] | 83 | 5375659901 | 4 | 16 | | 10 | [Len=31] | 3911 | [Len=28] | 2 | 4 | | 11 | [Len=34] | 401 | [Len=26] | 3 | 8 | | 12 | [Len=44] | 61 | [Len=19] | 4 | 16 | | 13 | [Len=50] |66935021479 | [Len=15] | 4 | 16 | | 14 | [Len=53] | 199 | [Len=21] | 5 | 32 | | 15 | [Len=60] | 73 | [Len=31] | 5 | 32 | | 16 | [Len=70] | 152249 | [Len=49] | 3 | 8 | | 17 | [Len=81] | 379 | [Len=49] | 5 | 32 | | 18 | [Len=84] | 5014363 | [Len=64] | 3 | 8 | | 19 | [Len=95] | 181 | [Len=72] | 4 | 16 | | 20 | [Len=102] | 191 | [Len=49] | 6 | 64 | | 21 | [Len=106] | 277 | [Len=57] | 5 | 32 | | 22 | [Len=117] | 155893 | [Len=79] | 4 | 16 | | 23 | [Len=125] | 35851 | [Len=73] | 5 | 32 | | 24 | [Len=137] | 431 | [Len=80] | 6 | 64 | | 25 | [Len=152] | 1499 | [Len=44] | 9 | 512 | | 26 | [Len=160] | [Len=160] | [Len=160] | 1 | 2 | +----+------------+------------+------------+------------+------------+ Len = number of digits in decimal representation (A055642), e.g. A103855(7) = 17!-17#+1 = 355687427585491 with Len=15. ----------- ------------------- page 3 2! - 2# + 1 n = 1: prime(1) = 2 ----------- ------------------- 2! = 2; 2# = 2 2! - 2# + 1 = 1 omega = 0; Omega = 0; mu = +1; tau = 1 sigma = 1; phi = 1 ======================================================================= ----------- ------------------- 3! - 3# + 1 n = 2: prime(2) = 3 ----------- ------------------- 3! = 6; 3# = 6 3! - 3# + 1 = 1 omega = 0; Omega = 0; mu = +1; tau = 1 sigma = 1; phi = 1 ======================================================================= ----------- ------------------- 5! - 5# + 1 n = 3: prime(3) = 5 ----------- ------------------- 5! = 120; 5# = 30 5! - 5# + 1 = 91 Factorization #1: 7 #2: 13 omega = 2; Omega = 2; mu = +1; tau = 4 sigma = 112; phi = 72 ======================================================================= ----------- ------------------- page 4 7! - 7# + 1 n = 4: prime(4) = 7 ----------- ------------------- 7! = 5040; 7# = 210 7! - 7# + 1 = 4831 Factorization 7! - 7# + 1 is prime omega = 1; Omega = 1; mu = -1; tau = 2 sigma = 4832; phi = 4830 ======================================================================= ------------- -------------------- 11! - 11# + 1 n = 5: prime(5) = 11 ------------- -------------------- 11! = 39 916800; 11# = 2310 11! - 11# + 1 = 39 914491 Factorization #1: 3673 #2: 10867 omega = 2; Omega = 2; mu = +1; tau = 4 sigma = 39 929032; phi = 39 899952 ======================================================================= ------------- -------------------- 13! - 13# + 1 n = 6: prime(6) = 13 ------------- -------------------- 13! = 6227 020800; 13# = 30030 13! - 13# + 1 = 6226 990771 Factorization #1: 16349 #2: 380879 omega = 2; Omega = 2; mu = +1; tau = 4 sigma = 6227 388000; phi = 6226 593544 ======================================================================= ------------- -------------------- page 5 17! - 17# + 1 n = 7: prime(7) = 17 ------------- -------------------- 17! = 355 687428 096000; 17# = 510510 [Len=15] 17! - 17# + 1 = 355 687427 585491 [Len=15] Factorization #1: 5 240507 #2: 67 872713 omega = 2; Omega = 2; mu = +1; tau = 4 sigma = 355 687500 698712; phi = 355 687354 472272 [Len=15] [Len=15] ======================================================================= ------------- -------------------- 19! - 19# + 1 n = 8: prime(8) = 19 ------------- -------------------- 19! = 121645 100408 832000; 19# = 9 699690 [Len=18] 19! - 19# + 1 = 121645 100399 132311 [Len=18] Factorization #1: 4159 #2: 29 248641 596329 [Len=14] omega = 2; Omega = 2; mu = +1; tau = 4 sigma = 121674 349040 732800; phi = 121615 851757 531824 [Len=18] [Len=18] ======================================================================= ------------- -------------------- page 6 23! - 23# + 1 n = 9: prime(9) = 23 ------------- -------------------- 23! = 25852 016738 884976 640000; 23# = 223 092870 [Len=23] 23! - 23# + 1 = 25852 016738 884753 547131 [Len=23] Factorization #1: 83 #2: 641 #3: 90 391277 #4: 5375 659901 [Len=10] omega = 4; Omega = 4; mu = +1; tau = 16 sigma = 26204 303786 955547 121568; phi = 25500 701524 255140 352000 [Len=23] [Len=23] ======================================================================= ------------- ---------------------- 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 = 8 841761 993739 701954 537146 306771 [Len=31] Factorization #1: 3911 #2: 2260 742008 115495 258127 626261 [Len=28] omega = 2; Omega = 2; mu = +1; tau = 4 sigma = 8 844022 735747 817449 795273 936944 [Len=31] phi = 8 839501 251731 586459 279018 676600 [Len=31] ======================================================================= ------------- ---------------------- 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 = 8222 838654 177922 817725 362319 509871 [Len=34] Factorization #1: 401 #2: 521743 #3: 39 302553 278734 052383 226897 [Len=26] omega = 3; Omega = 3; mu = -1; tau = 8 sigma = 8243 360285 859647 409507 002537 385024 [Len=34] phi = 8202 317101 101304 783411 826869 132800 [Len=34] ======================================================================= ------------- ---------------------- 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 = 13 763753 091226 345046 315979 581573 481661 865191 [Len=44] Factorization #1: 61 #2: 8 215303 #3: 12471 578198 147239 #4: 2 202226 772995 090243 [Len=19] omega = 4; Omega = 4; mu = +1; tau = 16 sigma = 13 989390 090648 918627 329493 090109 449081 402880 [Len=44] phi = 13 538116 146734 258320 615004 890482 768163 319520 [Len=44] ======================================================================= ------------- ---------------------- 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 = 33 452526 613163 807108 170062 053440 751360 901736 472791 [Len=50] Factorization #1: 66935 021479 #2: 83015 213929 #3: 16 542238 328879 #4: 363 934772 650919 [Len=15] omega = 4; Omega = 4; mu = +1; tau = 16 sigma = 33 452526 614068 666040 537953 433982 904696 696861 440000 [Len=50] phi = 33 452526 612258 948175 814325 380737 291579 993833 222336 [Len=50] ======================================================================= ------------- ---------------------- 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 = 60415 263063 373835 637355 132068 513997 494181 750668 329971 [Len=53] Factorization #1: 199 #2: 29759 #3: 16465 054861 #4: 4951 662085 741337 #5: 125 129894 757902 719183 [Len=21] omega = 5; Omega = 5; mu = -1; tau = 32 sigma = 60720 897706 557727 210385 404529 186919 928808 362487 808000 [Len=53] phi = 60109 648823 754522 119953 157281 254737 769950 623056 372480 [Len=53] ======================================================================= ------------- ---------------------- 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 = 258623 241511 168180 642964 355153 611979 969197 017499 337411 508591 [Len=60] Factorization #1: 73 #2: 30137 #3: 129 779047 #4: 680 584088 568619 #5: 1 330939 456978 870860 802004 244587 [Len=31] omega = 5; Omega = 5; mu = -1; tau = 32 sigma = 262174 726803 062868 033024 800139 064198 621485 440431 521941 506560 [Len=60] phi = 255071 991385 933787 902603 156926 156307 040878 480983 516964 071936 [Len=60] ======================================================================= ------------- --------------------- 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 = 4274 883284 060025 564298 013753 389399 649690 343788 366781 135513 522809 955271 [Len=70] Factorization #1: 152249 #2: 4054 792765 429333 #3: 6 924702 980663 032920 454126 415654 687339 896898 226563 [Len=49] omega = 3; Omega = 3; mu = -1; tau = 8 sigma = 4274 911362 295575 357988 336170 550379 218171 511135 106200 685953 037420 246000 [Len=70] phi = 4274 855205 824475 770621 540742 189746 147050 084694 458670 967863 387527 280832 [Len=70] ======================================================================= ------------- ---------------------- 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 = 138 683118 545689 835737 939019 720389 406345 902876 772687 432540 819372 179809 845787 360931 [Len=81] Factorization #1: 379 #2: 672577 #3: 589 957217 #4: 882 971248 154627 #5: 1 044420 282062 149924 430821 104235 868164 223697 278323 [Len=49] omega = 5; Omega = 5; mu = -1; tau = 32 sigma = 139 049244 040669 935693 334854 393863 914527 300955 773603 175914 147379 816101 666787 357440 [Len=81] phi = 138 316994 140059 853431 014952 460288 608362 915856 068489 102498 426115 440385 333655 289856 [Len=81] ======================================================================= ------------- ---------------------- 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 = 507580 213877 224798 800856 812176 625227 226004 528988 036003 099405 822192 604240 593029 016731 [Len=84] Factorization #1: 5 014363 #2: 10 671055 507103 #3: 9485 965377 776267 396881 274763 894939 726104 290993 687941 843639 154079 [Len=64] omega = 3; Omega = 3; mu = -1; tau = 8 sigma = 507580 315102 535449 591980 433197 351758 722447 346633 025606 629366 117700 113700 402353 172480 [Len=84] phi = 507580 112651 914148 028705 121911 451230 523324 260870 836279 021654 108672 470685 813104 212072 [Len=84] ======================================================================= ------------- ---------------------- 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 = 36471 110918 188685 288249 859096 605464 427167 635314 049524 593701 628500 267954 578622 320919 732944 120911 [Len=95] Factorization #1: 181 #2: 487 #3: 2 189153 #4: 189001 539040 224120 182266 849032 891902 263578 759062 151384 306450 949491 338546 012044 637621 [Len=72] omega = 4; Omega = 4; mu = +1; tau = 16 sigma = 36747 928653 013513 075799 425747 770371 842454 044290 691366 817753 343852 339366 997946 602341 699218 803008 [Len=95] phi = 36195 120942 442302 705946 455748 987789 124637 839802 011178 848462 979821 645874 304230 372780 970394 035200 [Len=95] ======================================================================= ------------- ---------------------- 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 = 850478 588567 862317 521167 644239 926010 288584 608120 796235 886430 763388 588680 377521 076867 153301 039032 584611 [Len=102] Factorization #1: 191 #2: 11863 #3: 13 731143 #4: 204 820009 #5: 101 013999 521489 443948 474737 551843 #6: 1 321219 117225 686113 328477 098882 653065 922629 498887 [Len=49] omega = 6; Omega = 6; mu = +1; tau = 64 sigma = 855003 489534 253126 969428 536609 231731 141434 595263 294879 994405 776122 634242 679550 096089 218314 753667 235840 [Len=102] phi = 845954 439003 045376 127380 218239 942116 796825 024017 050868 785926 880923 421745 829037 402657 321761 362959 048960 [Len=102] ======================================================================= ------------- ---------------------- 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 = 4470 115461 512684 340891 257138 125051 110076 800700 282905 015819 080092 370422 104067 142587 336304 430975 849378 676531 [Len=106] Factorization #1: 277 #2: 376803 030479 #3: 197053 025392 656137 #4: 475745 074227 968777 #5: 456 843213 172898 574585 091492 586868 694087 856701 992296 949593 [Len=57] omega = 5; Omega = 5; mu = -1; tau = 32 sigma = 4486 253062 468679 582254 463521 745890 029451 853529 954868 549309 821546 013481 294450 532035 169466 899453 685875 383040 [Len=106] phi = 4453 977860 556774 755126 749996 580347 652307 091016 231780 765082 880707 508724 482505 284231 403443 613104 841067 073536 [Len=106] ======================================================================= ------------- ---------------------- 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 = 94 618213 078297 528685 144171 539831 652069 808216 779571 907213 868063 227837 990693 501860 533358 593196 242835 327092 100915 445871 [Len=117] Factorization #1: 155893 #2: 44 155063 #3: 62 765353 836416 195481 988673 #4: 2 070668 756256 734773 435730 857278 252082 900254 900746 928402 198890 310050 277320 686853 [Len=79] omega = 4; Omega = 4; mu = +1; tau = 16 sigma = 894 623972 007530 194271 131248 269858 704865 498083 192819 963740 151070 990489 974967 981555 187895 955048 283924 194543 942864 396736 [Len=117] phi = 894 612454 149324 795613 486026 409775 602095 300698 630135 065950 775080 894068 944892 805667 506588 150009 859399 527051 503178 725376 [Len=117] ======================================================================= ------------- ---------------------- 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 = 39455 239697 206586 511897 471180 120610 571436 503407 643446 275224 357528 369751 562996 629334 879591 939836 706355 217604 148644 375982 007211 [Len=125] Factorization #1: 35851 #2: 12 708861 697663 #3: 17 249384 120083 #4: 639 348008 412700 777627 #5: 7 852100 479196 040972 894277 803807 353463 588745 397519 147818 007017 801639 508767 [Len=73] omega = 5; Omega = 5; mu = -1; tau = 32 sigma = 39456 340231 079854 106608 831074 515638 163877 475500 215205 624025 960253 160975 218165 520334 569180 310530 829259 056738 084210 135939 678208 [Len=125] phi = 39454 139163 333319 217980 372519 405141 901013 471907 635851 891039 234665 641828 566619 666124 054223 295841 127308 502427 816581 408703 002400 [Len=125] ======================================================================= ------------- ---------------------- 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 = 16507 955160 908461 081216 919262 453619 309839 666236 496541 854913 520707 833171 034378 509739 399912 570787 600662 705311 641103 411249 229349 462398 641691 [Len=137] Factorization #1: 431 #2: 1 421867 #3: 8 097071 #4: 7414 443082 592873 #5: 27 388176 364546 578906 711671 #6: 16 382773 866485 664161 013567 326480 632837 456842 781188 331023 568762 736257 503952 578431 [Len=80] omega = 6; Omega = 6; mu = +1; tau = 64 sigma = 16546 270361 484738 711757 857867 670213 354001 679050 361770 550081 500956 131224 538312 127118 804830 006145 399504 285286 675980 805980 334053 855500 173312 [Len=137] phi = 16469 640023 670606 377457 840207 477800 262205 774032 762309 301273 509477 081941 077454 888099 286306 730236 251816 204591 416781 451803 828682 806905 120000 [Len=137] ======================================================================= ------------- ---------------------- 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 = 96 192759 682482 119853 328425 949563 698712 343813 919172 976158 104477 319333 745612 481875 498805 879175 589072 651261 284189 679675 862079 103886 801575 246897 852668 243931 [Len=152] Factorization #1: 1499 #2: 7079 #3: 335477 #4: 158510 733961 #5: 14 256662 806021 #6: 10 133442 888029 683961 #7: 11 095026 582905 344111 445081 #8: 5869 610044 290131 155677 429733 #9: 18 119028 712301 521338 946889 484307 215249 548651 [Len=44] omega = 9; Omega = 9; mu = -1; tau = 512 sigma = 96 270815 468790 194231 484793 950920 842530 582782 010168 668481 825355 128495 660330 960644 764612 788032 587412 804192 452194 333288 946419 269186 991250 592870 942100 480000 [Len=152] phi = 96 114722 489795 335599 667561 779975 411090 681475 831400 638138 079874 218026 034304 267302 815176 484595 607130 822773 693351 340690 633524 954551 291860 049924 861952 000000 [Len=152] ======================================================================= --------------- ----------------------- 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 = 9425 947759 838359 420851 623124 482936 749562 312794 702543 768327 889353 416977 599316 221476 503087 861591 808346 911623 490003 549599 583136 843938 244766 639099 936683 119492 636931 [Len=160] Factorization 101! - 101# + 1 is prime omega = 1; Omega = 1; mu = -1; tau = 2 sigma = 9425 947759 838359 420851 623124 482936 749562 312794 702543 768327 889353 416977 599316 221476 503087 861591 808346 911623 490003 549599 583136 843938 244766 639099 936683 119492 636932 [Len=160] phi = 9425 947759 838359 420851 623124 482936 749562 312794 702543 768327 889353 416977 599316 221476 503087 861591 808346 911623 490003 549599 583136 843938 244766 639099 936683 119492 636930 [Len=160] =======================================================================