+-----------------------------------------------------+ | A104350: partial products of largest prime factors | | and factorizations of A104350(n)-1 and A104350(n)+1 | | for n = 1 to 50 | +----------------------------------------------------- Contents: page Table 1: GPP = Partial products of largest prime factors 2 Table 2: tau, sigma, and phi for GPP 3 Table 3: Smallest and greatest prime factor of GPP-1 and GPP+1; number of distinct factors 4 Table 4: tau, sigma, and phi for GPP-1 and GPP+1 5 Table 5: Primes in GPP-1 and in GPP+1; twin primes 6 | n = 1,2,3 7 | n = 4, 5 8 | n = 6, 7 9 | n = 8, 9 10 | n = 10, 11 11 | n = 12, 13 12 | n = 14, 15 13 | n = 16, 17 14 | n = 18, 19 15 | n = 20, 21 16 | n = 22, 23 17 | n = 24, 25 18 | n = 26, 27 19 | n = 28, 29 20 Factorizations | n = 30 21 and details: | n = 31 22 | n = 32 23 | n = 33 24 | n = 34 25 | n = 35 26 | n = 36 27 | n = 37 28 | n = 38 29 | n = 39 30 | n = 40 31 | n = 41 32 | n = 42 33 | n = 43 34 | n = 44 35 | n = 45 36 | n = 46 37 | n = 47 38 | n = 48 39 | n = 49 40 | n = 50 41 Appendix: Primes in factorizations 42 All factorizations by using Dario Alpern's ECM-program http://www.alpertron.com.ar/ECM.HTM A104350.txt Reinhard Zumkeller ([email protected]), Mar 04 2005 page 2 +----+----------------------------------------------------+---------+ | | Partial product of largest prime factors | #digits | | n +----------------------------------------------------+---------+ | | A104350 | A104351 | +----+----------------------------------------------------+---------+ | 1 | 1 | 1 | | 2 | 2 | 1 | | 3 | 6 | 1 | | 4 | 12 | 2 | | 5 | 60 | 2 | | 6 | 180 | 3 | | 7 | 1260 | 4 | | 8 | 2520 | 4 | | 9 | 7560 | 4 | | 10 | 37800 | 5 | | 11 | 415800 | 6 | | 12 | 1 247400 | 7 | | 13 | 16 216200 | 8 | | 14 | 113 513400 | 9 | | 15 | 567 567000 | 9 | | 16 | 1135 134000 | 10 | | 17 | 19297 278000 | 11 | | 18 | 57891 834000 | 11 | | 19 | 1 099944 846000 | 13 | | 20 | 5 499724 230000 | 13 | | 21 | 38 498069 610000 | 14 | | 22 | 423 478765 710000 | 15 | | 23 | 9740 011611 330000 | 16 | | 24 | 29220 034833 990000 | 17 | | 25 | 146100 174169 950000 | 18 | | 26 | 1 899302 264209 350000 | 19 | | 27 | 5 697906 792628 050000 | 19 | | 28 | 39 885347 548396 350000 | 20 | | 29 | 1156 675078 903494 150000 | 22 | | 30 | 5783 375394 517470 750000 | 22 | | 31 | 179284 637230 041593 250000 | 24 | | 32 | 358569 274460 083186 500000 | 24 | | 33 | 3 944262 019060 915051 500000 | 25 | | 34 | 67 052454 324035 555875 500000 | 26 | | 35 | 469 367180 268248 891128 500000 | 27 | | 36 | 1408 101540 804746 673385 500000 | 28 | | 37 | 52099 757009 775626 915263 500000 | 29 | | 38 | 989895 383185 736911 390006 500000 | 30 | | 39 | 12 868639 981414 579848 070084 500000 | 32 | | 40 | 64 343199 907072 899240 350422 500000 | 32 | | 41 | 2638 071196 189988 868854 367322 500000 | 34 | | 42 | 18466 498373 329922 081980 571257 500000 | 35 | | 43 | 794059 430053 186649 525164 564072 500000 | 36 | | 44 | 8 734653 730585 053144 776810 204797 500000 | 37 | | 45 | 43 673268 652925 265723 884051 023987 500000 | 38 | | 46 | 1004 485179 017281 111649 333173 551712 500000 | 40 | | 47 | 47210 803413 812212 247518 659156 930487 500000 | 41 | | 48 | 141632 410241 436636 742555 977470 791462 500000 | 42 | | 49 | 991426 871690 056457 197891 842295 540237 500000 | 42 | | 50 | 4 957134 358450 282285 989459 211477 701187 500000 | 43 | +----+----------------------------------------------------+---------+ page 3 x = A104350(n) +------------+-------------------------+-------------------------+ | tau(x) | sigma(x) | phi(x) | +----+------------+-------------------------+-------------------------+ | n | A104352 | A104353 | A104354 | +----+------------+-------------------------+-------------------------+ | 1 | 1 | 1 | 1 | | 2 | 2 | 3 | 1 | | 3 | 4 | 12 | 2 | | 4 | 6 | 28 | 4 | | 5 | 12 | 168 | 16 | | 6 | 18 | 546 | 48 | | 7 | 36 | 4368 | 288 | | 8 | 48 | 9360 | 576 | | 9 | 64 | 28800 | 1728 | | 10 | 96 | 148800 | 8640 | | 11 | 192 | 1 785600 | 86400 | | 12 | 240 | 5 401440 | 259200 | | 13 | 480 | 75 620160 | 3 110400 | | 14 | 720 | 538 793640 | 21 772800 | | 15 | 960 | 2711 348640 | 108 864000 | | 16 | 1200 | 5603 453856 | 217 728000 | | 17 | 2400 | 100862 169408 | 3483 648000 | | 18 | 2880 | 303420 079872 | 10450 944000 | | 19 | 5760 | 6 068401 597440 | 188116 992000 | | 20 | 7200 | 30 380907 997440 | 940584 960000 | | 21 | 9600 | 213 199354 368000 | 6 584094 720000 | | 22 | 14400 | 2362 959510 912000 | 72 425041 920000 | | 23 | 28800 | 56711 028261 888000 | 1593 350922 240000 | | 24 | 33600 | 170288 884313 856000 | 4780 052766 720000 | | 25 | 40320 | 851662 461113 856000 | 23900 263833 600000 | | 26 | 60480 | 11 132445 027416 832000 | 310703 429836 800000 | | 27 | 69120 | 33 407520 301854 720000 | 932110 289510 400000 | | 28 | 86400 |233 936160 913737 676800 | 6 524772 026572 800000 | | 29 | 172800 | [L=22] |182 693616 744038 400000 | | 30 | 201600 | [L=23] |913 468083 720192 000000 | | 31 | 403200 | [L=25] | [L=23] | | 32 | 483840 | [L=25] | [L=23] | | 33 | 645120 | [L=26] | [L=24] | | 34 | 967680 | [L=27] | [L=26] | | 35 | 1 161216 | [L=28] | [L=26] | | 36 | 1 306368 | [L=28] | [L=27] | | 37 | 2 612736 | [L=30] | [L=28] | | 38 | 3 919104 | [L=31] | [L=30] | | 39 | 5 225472 | [L=32] | [L=30] | | 40 | 5 971968 | [L=33] | [L=31] | | 41 | 11 943936 | [L=35] | [L=33] | | 42 | 13 934592 | [L=36] | [L=34] | | 43 | 27 869184 | [L=37] | [L=36] | | 44 | 34 836480 | [L=38] | [L=37] | | 45 | 39 191040 | [L=39] | [L=37] | | 46 | 58 786560 | [L=40] | [L=39] | | 47 | 117 573120 | [L=42] | [L=40] | | 48 | 130 636800 | [L=43] | [L=41] | | 49 | 149 299200 | [L=43] | [L=42] | | 50 | 165 888000 | [L=44] | [L=42] | +----+------------+-------------------------+-------------------------+ page 4 x = A104350(n) - 1 x = A104350(n) + 1 +---------+---------+---------++---------+---------+---------+ | MinPrFct(x) | || MinPrFct(x) | | | MaxPrFct(x) || MaxPrFct(x) | | | omega(x) || | omega(x) | +----++---------+---------+---------++---------+---------+---------+ | n || A104358 | A104359 | A104360 || A104366 | A104367 | A104368 | +----++---------+---------+---------++---------+---------+---------+ | 1 || | | || 2 | 2 | 1 | | 2 || 1 | 1 | 0 || 3 | 3 | 1 | | 3 || 5 | 5 | 1 || 7 | 7 | 1 | | 4 || 11 | 11 | 1 || 13 | 13 | 1 | | 5 || 59 | 59 | 1 || 61 | 61 | 1 | | 6 || 178 | 179 | 1 || 181 | 181 | 1 | | 7 || 1259 | 1259 | 1 || 13 | 97 | 2 | | 8 || 11 | 229 | 2 || 2521 | 2521 | 1 | | 9 || 7559 | 7559 | 1 || 7561 | 7561 | 1 | | 10 || 37799 | 37799 | 1 || 103 | 367 | 2 | | 11 || 415799 | 415799 | 1 || 415801 | 415801 | 1 | | 12 || 71 | 17569 | 2 || 1247401 | 1247401 | 1 | | 13 || 227 | 71437 | 2 || 167 | 97103 | 2 | | 14 || 5981 | 18979 | 2 || 191 | 594311 | 2 | | 15 || 9067 | 62597 | 2 || 211 | 2689891 | 2 | | 16 || [L=10] | [L=10] | 1 || 127 | 269 | 4 | | 17 || 11717 | 1646947 | 2 || 23 | 415147 | 4 | | 18 || 61 | 445771 | 3 || 40357 | 1434493 | 2 | | 19 || 79 | 277021 | 3 || [L=13] | [L=13] | 1 | | 20 || [L=13] | [L=13] | 1 || 349 | 13421 | 4 | | 21 || 97 | [L=10] | 3 || 41 | [L=12] | 2 | | 22 || 1543 | 721381 | 3 || 251 | [L=13] | 2 | | 23 || 31 |46313123 | 3 || 37 | [L=13] | 3 | | 24 || [L=17] | [L=17] | 1 || 2243 | [L=14] | 2 | | 25 || 8937119 | [L=11] | 2 || [L=18] | [L=18] | 1 | | 26 || 181 | [L=10] | 4 || 103 | 1389833 | 4 | | 27 || 401 | 4939567 | 4 || 53 | [L=14] | 3 | | 28 || 124759 | [L=15] | 2 || 1217 | [L=10] | 3 | | 29 || 443851 | [L=16] | 2 || [L=22] | [L=22] | 1 | | 30 || 31 | [L=9] | 5 || 47 | [L=12] | 3 | | 31 || 2141 |15701603 | 4 || 2939 | [L=11] | 4 | | 32 || 3082663 | [L=18] | 2 || 251 |11159737 | 5 | | 33 || 8191 | [L=12] | 3 || 857 | [L=14] | 4 | | 34 ||37230797 | [L=19] | 2 || 41 | [L=13] | 5 | | 35 || 1697 | [L=18] | 3 || 547 | [L=14] | 4 | | 36 || [L=28] | [L=28] | 1 || 13127 | [L=15] | 3 | | 37 || 10613 | [L=18] | 3 || 47 | [L=21] | 3 | | 38 || 73 | [L=16] | 4 || 48563 | [L=10] | 4 | | 39 || 59 | [L=27] | 3 || 281 |16389203 | 7 | | 40 || 107 | [L=10] | 5 || [L=16] | [L=17] | 2 | | 41 || 79 | [L=30] | 3 || 479 | [L=13] | 5 | | 42 || 617 | [L=15] | 4 || 373 | [L=15] | 4 | | 43 || 163 | [L=20] | 4 || 2179 | [L=21] | 4 | | 44 || 173 | [L=19] | 5 || [L=18] | [L=20] | 2 | | 45 || [L=38] | [L=38] | 1 || 17491 | [L=14] | 4 | | 46 || 911 | [L=19] | 3 || 1399 | [L=19] | 3 | | 47 || 71 | [L=28] | 4 || 97 | [L=22] | 4 | | 48 || [L=16] | [L=26] | 2 ||22893547 | [L=34] | 2 | | 49 || [L=15] | [L=28] | 2 || 1069973 | [L=29] | 3 | | 50 || [L=13] | [L=30] | 2 || 61 | [L=20] | 7 | +----++---------+---------+---------++---------+---------+---------+ page 5 x = A104350(n) - 1 +------------+-------------------------+-------------------------+ | tau(x) | sigma(x) | phi(x) | +----+------------+-------------------------+-------------------------+ | n | A104361 | A104362 | A104363 | +----+------------+-------------------------+-------------------------+ | 1 | | | | | 2 | 1 | 1 | 1 | | 3 | 2 | 6 | 4 | | 4 | 2 | 12 | 4 | | 5 | 2 | 60 | 58 | | 6 | 2 | 180 | 178 | | 7 | 2 | 1260 | 1258 | | 8 | 4 | 2760 | 2280 | | 9 | 2 | 7560 | 7558 | | 10 | 2 | 37800 | 37798 | | 11 | 2 | 415800 | 415798 | | 12 | 4 | 1 265040 | 1 229760 | | 13 | 4 | 16 287864 | 16 144536 | | 14 | 4 | 113 538360 | 113 488440 | | 15 | 4 | 567 638664 | 567 495336 | | 16 | 2 | 1135 134000 | 1135 133998 | | 17 | 4 | 19298 936664 | 19295 619336 | | 18 | 8 | 58868 650320 | 56915 913600 | | 19 | 8 | 1 113894 381120 | 1 085995 965600 | | 20 | 2 | 5 499724 230000 | 5 499724 229998 | +----+------------+-------------------------+-------------------------+ x = A104350(n) + 1 +------------+-------------------------+-------------------------+ | tau(x) | sigma(x) | phi(x) | +----+------------+-------------------------+-------------------------+ | n | A104369 | A104370 | A104371 | +----+------------+-------------------------+-------------------------+ | 1 | 2 | 3 | 1 | | 2 | 2 | 4 | 2 | | 3 | 2 | 8 | 6 | | 4 | 2 | 14 | 12 | | 5 | 2 | 62 | 60 | | 6 | 2 | 182 | 180 | | 7 | 4 | 1372 | 1152 | | 8 | 2 | 2522 | 2520 | | 9 | 2 | 7562 | 7560 | | 10 | 4 | 38272 | 37332 | | 11 | 2 | 415802 | 415800 | | 12 | 2 | 1 247402 | 1 247400 | | 13 | 4 | 16 313472 | 16 118932 | | 14 | 4 | 114 107904 | 112 918900 | | 15 | 4 | 570 257104 | 564 876900 | | 16 | 16 | 1161 216000 | 1109 481408 | | 17 | 16 | 21043 021824 | 17645 365584 | | 18 | 4 | 57893 308852 | 57890 359152 | | 19 | 2 | 1 099944 846002 | 1 099944 846000 | | 20 | 16 | 5 530978 026000 | 5 468570 553600 | +----+------------+-------------------------+-------------------------+ page 6 +----------------------------------------------+ | Primes of the form A104350(k)-1 | +----+----------------------------------------------+----+ | n | A104364(n) | k | +----+----------------------------------------------+----+ | 1 | 5 | 3 | | 2 | 11 | 4 | | 3 | 59 | 5 | | 4 | 179 | 6 | | 5 | 1259 | 7 | | 6 | 7559 | 9 | | 7 | 37799 | 10 | | 8 | 415799 | 11 | | 9 | 1135 133999 | 16 | | 10 | 5 499724 229999 | 20 | | 11 | 29220 034833 989999 | 24 | | 12 | 1408 101540 804746 673385 499999 | 36 | | 13 | 43 673268 652925 265723 884051 023987 499999 | 45 | +----+----------------------------------------------+----+ +----------------------------------------------+ | Primes of the form A104350(k)+1 | +----+----------------------------------------------+----+ | n | A104372(n) | k | +----+----------------------------------------------+----+ | 1 | 2 | 1 | | 2 | 3 | 2 | | 3 | 7 | 3 | | 4 | 13 | 4 | | 5 | 61 | 5 | | 6 | 181 | 6 | | 7 | 2521 | 8 | | 8 | 7561 | 9 | | 9 | 415801 | 11 | | 10 | 1 247401 | 12 | | 11 | 1 099944 846001 | 19 | | 12 | 146100 174169 950001 | 25 | | 13 | 1156 675078 903494 150001 | 29 | +----+----------------------------------------------+----+ +--------------------------------------------+ | Twin prime pairs (A001359(i),A006512(i)) | | of the form (A104350(k)-1,A104350(k)+1) | +----+--------------+------------+------+---------+ | n | A104373(n)=k | A104350(k) | i | +----+--------------+------------+------+ | 1 | 3 | 6 | 2 | (5,7) | 2 | 4 | 12 | 3 | (11,13) | 3 | 5 | 60 | 7 | (59,61) | 4 | 6 | 180 | 13 | (179,181) | 5 | 9 | 7560 | 171 | (7559,7561) | 6 | 11 | 415800 | 3928 | (415799,415801) +----+--------------+------------+------+ ----- page 7 n = 1 ----- GPP = 1 omega = 0; Omega = 0; tau = 1; sigma = 1; phi = 1 ....................................................................... GPP + 1 = 2 is prime omega = 1; Omega = 1; tau = 2; sigma = 3; phi = 1 ======================================================================= ----- n = 2 greatest prime factor = 2 ----- GPP = 2 omega = 1; Omega = 1; tau = 2; sigma = 3; phi = 1 ....................................................................... GPP - 1 = 1 omega = 0; Omega = 0; tau = 1; sigma = 1; phi = 1 ....................................................................... GPP + 1 = 3 is prime omega = 1; Omega = 1; tau = 2; sigma = 4; phi = 2 ======================================================================= ----- n = 3 greatest prime factor = 3 ----- GPP = 2*3 = 6 omega = 2; Omega = 2; tau = 4; sigma = 12; phi = 2 ....................................................................... GPP - 1 = 5 is prime omega = 1; Omega = 1; tau = 2; sigma = 6; phi = 4 ....................................................................... GPP + 1 = 7 is prime omega = 1; Omega = 1; tau = 2; sigma = 8; phi = 6 ======================================================================= ----- page 8 n = 4 greatest prime factor = 2 ----- GPP = 2*3*2 = 12 omega = 2; Omega = 3; tau = 6; sigma = 28; phi = 4 ....................................................................... GPP - 1 = 11 is prime omega = 1; Omega = 1; tau = 2; sigma = 12; phi = 10 ....................................................................... GPP + 1 = 13 is prime omega = 1; Omega = 1; tau = 2; sigma = 14; phi = 12 ======================================================================= ----- n = 5 greatest prime factor = 5 ----- GPP = 2*3*2*5 = 60 omega = 3; Omega = 4; tau = 12; sigma = 168; phi = 16 ....................................................................... GPP - 1 = 59 is prime omega = 1; Omega = 1; tau = 2; sigma = 60; phi = 58 ....................................................................... GPP + 1 = 61 is prime omega = 1; Omega = 1; tau = 2; sigma = 62; phi = 60 ======================================================================= ----- page 9 n = 6 greatest prime factor = 3 ----- GPP = 2*3*2*5*3 = 180 omega = 3; Omega = 5; tau = 18; sigma = 546; phi = 48 ....................................................................... GPP - 1 = 179 is prime omega = 1; Omega = 1; tau = 2; sigma = 180; phi = 178 ....................................................................... GPP + 1 = 181 is prime omega = 1; Omega = 1; tau = 2; sigma = 182; phi = 180 ======================================================================= ----- n = 7 greatest prime factor = 7 ----- GPP = 2*3*2*5*3*7 = 1260 omega = 4; Omega = 6; tau = 36; sigma = 4368; phi = 288 ....................................................................... GPP - 1 = 1259 is prime omega = 1; Omega = 1; tau = 2; sigma = 1260; phi = 1258 ....................................................................... GPP + 1 = 1261 = 13 * 97 omega = 2; Omega = 2; tau = 4; sigma = 1372; phi = 1152 ======================================================================= ----- page 10 n = 8 greatest prime factor = 2 ----- GPP = 2*3*2*5*3*7*2 = 2520 omega = 4; Omega = 7; tau = 48; sigma = 9360; phi = 576 ....................................................................... GPP - 1 = 2519 = 11 * 229 omega = 2; Omega = 2; tau = 4; sigma = 2760; phi = 2280 ....................................................................... GPP + 1 = 2521 is prime omega = 1; Omega = 1; tau = 2; sigma = 2522; phi = 2520 ======================================================================= ----- n = 9 greatest prime factor = 3 ----- GPP = 2*3*2*5*3*7*2*3 = 7560 omega = 4; Omega = 8; tau = 64; sigma = 28800; phi = 1728 ....................................................................... GPP - 1 = 7559 is prime omega = 1; Omega = 1; tau = 2; sigma = 7560; phi = 7558 ....................................................................... GPP + 1 = 7561 is prime omega = 1; Omega = 1; tau = 2; sigma = 7562; phi = 7560 ======================================================================= ------ page 11 n = 10 greatest prime factor = 5 ------ GPP = 2*3*2*5*3*7*2*3*5 = 37800 omega = 4; Omega = 9; tau = 96; sigma = 148800; phi = 8640 ....................................................................... GPP - 1 = 37799 is prime omega = 1; Omega = 1; tau = 2; sigma = 37800; phi = 37798 ....................................................................... GPP + 1 = 37801 = 103 * 367 omega = 2; Omega = 2; tau = 4; sigma = 38272; phi = 37332 ======================================================================= ------ n = 11 greatest prime factor = 11 ------ GPP = 2*3*2*5*3*7*2*3*5*11 = 415800 omega = 5; Omega = 10; tau = 192; sigma = 1 785600; phi = 86400 ....................................................................... GPP - 1 = 415799 is prime omega = 1; Omega = 1; tau = 2; sigma = 415800; phi = 415798 ....................................................................... GPP + 1 = 415801 is prime omega = 1; Omega = 1; tau = 2; sigma = 415802; phi = 415800 ======================================================================= ------ page 12 n = 12 greatest prime factor = 3 ------ GPP = 2*3*2*5*3*7*2*3*5*11*3 = 1247400 omega = 5; Omega = 11; tau = 240; sigma = 5401440; phi = 259200 ....................................................................... GPP - 1 = 1247399 = 71 * 17569 omega = 2; Omega = 2; tau = 4; sigma = 1265040; phi = 1229760 ....................................................................... GPP + 1 = 1247401 is prime omega = 1; Omega = 1; tau = 2; sigma = 1 247402; phi = 1 247400 ======================================================================= ------ n = 13 greatest prime factor = 13 ------ GPP = 2*3*2*5*3*7*2*3*5*11*3*13 = 16216200 omega = 6; Omega = 12; tau = 480; sigma = 75 620160; phi = 3 110400 ....................................................................... GPP - 1 = 16216199 = 227 * 71437 omega = 2; Omega = 2; tau = 4; sigma = 16 287864; phi = 16 144536 ....................................................................... GPP + 1 = 16216201 = 167 * 97103 omega = 2; Omega = 2; tau = 4; sigma = 16 313472; phi = 16 118932 ======================================================================= ------ page 13 n = 14 greatest prime factor = 7 ------ GPP = 2*3*2*5*3*7*2*3*5*11*3*13*7 = 113513400 omega = 6; Omega = 13; tau = 720; sigma = 538793640; phi = 21772800 ....................................................................... GPP - 1 = 113513399 = 5981 * 18979 omega = 2; Omega = 2; tau = 4; sigma = 113 538360; phi = 113 488440 ....................................................................... GPP + 1 = 113513401 = 191 * 594311 omega = 2; Omega = 2; tau = 4; sigma = 114 107904; phi = 112 918900 ======================================================================= ------ n = 15 greatest prime factor = 5 ------ GPP = 2*3*2*5*3*7*2*3*5*11*3*13*7*5 = 567 567000 omega = 6; Omega = 14; tau = 960 sigma = 2711 348640; phi = 108 864000 ....................................................................... GPP - 1 = 567 566999 = 9067 * 62597 omega = 2; Omega = 2; tau = 4 sigma = 567 638664; phi = 567 495336 ....................................................................... GPP + 1 = 567 567001 = 211 * 2 689891 omega = 2; Omega = 2; tau = 4 sigma = 570 257104; phi = 564 876900 ======================================================================= ------ page 14 n = 16 greatest prime factor = 2 ------ GPP = 2*3*2*5*3*7*2*3*5*11*3*13*7*5*2 = 1135 134000 omega = 6; Omega = 15; tau = 1200 sigma = 5603 453856; phi = 217 728000 ....................................................................... GPP - 1 = 1135 133999 is prime omega = 1; Omega = 1; tau = 2 sigma = 1135 134000; phi = 1135 133998 ....................................................................... GPP + 1 = 1135 134001 = 127 * 149 * 223 * 269 omega = 4; Omega = 4; tau = 16 sigma = 1161 216000; phi = 1109 481408 ======================================================================= ------ n = 17 greatest prime factor = 17 ------ GPP = 2*3*2*5*3*7*2*3*5*11*3*13*7*5*2*17 = 19297 278000 omega = 7; Omega = 16; tau = 2400 sigma = 100862 169408; phi = 3483 648000 ....................................................................... GPP - 1 = 19297 277999 = 11717 * 1 646947 omega = 2; Omega = 2; tau = 4 sigma = 19298 936664; phi = 19295 619336 ....................................................................... GPP + 1 = 19297 278001 = 23 * 43 * 47 * 415147 omega = 4; Omega = 4; tau = 16 sigma = 21043 021824; phi = 17645 365584 ======================================================================= ------ page 15 n = 18 greatest prime factor = 3 ------ GPP = 2*3*2*5*3*7*2*3*5*11*3*13*7*5*2*17*3 = 57891 834000 omega = 7; Omega = 17; tau = 2880 sigma = 303420 079872; phi = 10450 944000 ....................................................................... GPP - 1 = 57891 833999 = 61 * 2129 * 445771 omega = 3; Omega = 3; tau = 8 sigma = 58868 650320; phi = 56915 913600 ....................................................................... GPP + 1 = 57891 834001 = 40357 * 1 434493 omega = 2; Omega = 2; tau = 4 sigma = 57893 308852; phi = 57890 359152 ======================================================================= ------ n = 19 greatest prime factor = 19 ------ GPP = 2*3*2*5*3*7*2*3*5*11*3*13*7*5*2*17*3*19 = 1 099944 846000 omega = 8; Omega = 18; tau = 5760 sigma = 6 068401 597440; phi = 188116 992000 ....................................................................... GPP - 1 = 1 099944 845999 = 79 * 50261 * 277021 omega = 3; Omega = 3; tau = 8 sigma = 1 113894 381120; phi = 1 085995 965600 ....................................................................... GPP + 1 = 1 099944 846001 is prime omega = 1; Omega = 1; tau = 2 sigma = 1 099944 846002; phi = 1 099944 846000 ======================================================================= ------ page 16 n = 20 greatest prime factor = 5 ------ GPP = 2*3*2*5*3*7*2*3*5*11*3*13*7*5*2*17*3*19*5 = 5 499724 230000 omega = 8; Omega = 19; tau = 7200 sigma = 30 380907 997440; phi = 940584 960000 ....................................................................... GPP - 1 = 5 499724 229999 is prime omega = 1; Omega = 1; tau = 2 sigma = 5 499724 230000; phi = 5 499724 229998 ....................................................................... GPP + 1 = 5 499724 230001 = 349 * 421 * 2789 * 13421 omega = 4; Omega = 4; tau = 16 sigma = 5 530978 026000; phi = 5 468570 553600 ======================================================================= ------ n = 21 greatest prime factor = 7 ------ GPP = 2*3*2*5*3*7*2*3*5*11*3*13*7*5*2*17*3*19*5*7 = 38 498069 610000 omega = 8; Omega = 20; tau = 9600 sigma = 213 199354 368000; phi = 6 584094 720000 ....................................................................... GPP - 1 = 38 498069 609999 = 97 * 179 * 2217 247573 omega = 3; Omega = 3; tau = 8 sigma = 39 112247 205360; phi = 37 888326 510336 ....................................................................... GPP + 1 = 38 498069 610001 = 41 * 938977 307561 omega = 2; Omega = 2; tau = 4 sigma = 39 437046 917604; phi = 37 559092 302400 ======================================================================= ------ page 17 n = 22 greatest prime factor = 11 ------ GPP = 2*3*2*5*3*7*2*3*5*11*3*13*7*5*2*17*3*19*5*7*11 = 423 478765 710000 omega = 8; Omega = 21; tau = 14400 sigma = 2362 959510 912000; phi = 72 425041 920000 ....................................................................... GPP - 1 = 423 478765 709999 = 1543 * 380453 * 721381 omega = 3; Omega = 3; tau = 8 sigma = 423 754918 508832; phi = 423 202615 117920 ....................................................................... GPP + 1 = 423 478765 710001 = 251 * 1 687166 397251 omega = 2; Omega = 2; tau = 4 sigma = 425 165932 107504; phi = 421 791599 312500 ======================================================================= ------ n = 23 greatest prime factor = 23 ------ GPP = 2*3*2*5*3*7*2*3*5*11*3*13*7*5*2*17*3*19*5*7*11*23 = 9740 011611 330000 omega = 9; Omega = 22; tau = 28800 sigma = 56711 028261 888000; phi = 1593 350922 240000 ....................................................................... GPP - 1 = 9740 011611 329999 = 31 * 6 784123 * 46 313123 omega = 3; Omega = 3; tau = 8 sigma = 10054 207233 388032; phi = 9425 816095 466520 ....................................................................... GPP + 1 = 9740 011611 330001 = 37 * 43 * 6 121943 187511 omega = 3; Omega = 3; tau = 8 sigma = 10235 889009 520064; phi = 9256 378099 515120 ======================================================================= ------ page 18 n = 24 greatest prime factor = 3 ------ GPP = 2*3*2*5*3*7*2*3*5*11*3*13*7*5*2*17*3*19*5*7*11*23*3 = 29220 034833 990000 omega = 9; Omega = 23; tau = 33600 sigma = 170288 884313 856000; phi = 4780 052766 720000 ....................................................................... GPP - 1 = 29220 034833 989999 is prime omega = 1; Omega = 1; tau = 2 sigma = 29220 034833 990000; phi = 29220 034833 989998 ....................................................................... GPP + 1 = 29220 034833 990001 = 2243 * 13 027211 250107 omega = 2; Omega = 2; tau = 4 sigma = 29233 062045 242352; phi = 29207 007622 737652 ======================================================================= ------ n = 25 greatest prime factor = 5 ------ GPP = 2*3*2*5*3*7*2*3*5*11*3*13*7*5*2*17*3*19*5*7*11*23*3*5 = 146100 174169 950000 omega = 9; Omega = 24; tau = 40320 sigma = 851662 461113 856000; phi = 23900 263833 600000 ....................................................................... GPP - 1 = 146100 174169 949999 = 8 937119 * 16347 569521 omega = 2; Omega = 2; tau = 4 sigma = 146100 190526 456640; phi = 146100 157813 443360 ....................................................................... GPP + 1 = 146100 174169 950001 is prime omega = 1; Omega = 1; tau = 2 sigma = 146100 174169 950002; phi = 146100 174169 950000 ======================================================================= ------ page 19 n = 26 greatest prime factor = 13 ------ GPP = 2*3*2*5*3*7*2*3*5*11*3*13*7*5*2*17*3*19*5*7*11*23*3*5*13 = 1 899302 264209 350000 omega = 9; Omega = 25; tau = 60480 sigma = 11 132445 027416 832000; phi = 310703 429836 800000 ....................................................................... GPP - 1 = 1 899302 264209 349999 = 181 * 331 * 5801 * 5464 930609 omega = 4; Omega = 4; tau = 16 sigma = 1 915895 635570 469280; phi = 1 882777 893068 160000 ....................................................................... GPP + 1 = 1 899302 264209 350001 = 103 * 19069 * 695771 * 1 389833 omega = 4; Omega = 4; tau = 16 sigma = 1 917846 796927 501440; phi = 1 880759 745476 519040 ======================================================================= ------ n = 27 greatest prime factor = 3 ------ GPP = 2*3*2*5*3*7*2*3*5*11*3*13*7*5*2*17*3*19*5*7*11*23*3*5*13*3 = 5 697906 792628 050000 omega = 9; Omega = 26; tau = 69120 sigma = 33 407520 301854 720000; phi = 932110 289510 400000 ....................................................................... GPP - 1 = 5 697906 792628 049999 = 401 * 37799 * 76103 * 4 939567 omega = 4; Omega = 4; tau = 16 sigma = 5 712343 370808 883200; phi = 5 683471 349506 454400 ....................................................................... GPP + 1 = 5 697906 792628 050001 = 53 * 10139 * 10 603380 543703 omega = 3; Omega = 3; tau = 8 sigma = 5 805987 050510 562240; phi = 5 589847 741506 645552 ======================================================================= ------ page 20 n = 28 greatest prime factor = 7 ------ GPP = 2*3*2*5*3*7*2*3*5*11*3*13*7*5*2*17*3*19*5*7*11*23*3*5*13*3*7 = 39 885347 548396 350000 omega = 9; Omega = 27; tau = 86400 sigma = 233 936160 913737 676800; phi = 6 524772 026572 800000 ....................................................................... GPP - 1 = 39 885347 548396 349999 = 124759 * 319 699160 368361 omega = 2; Omega = 2; tau = 4 sigma = 39 885667 247556 843120; phi = 39 885027 849235 856880 ....................................................................... GPP + 1 = 39 885347 548396 350001 = 1217 * 15 391787 * 2129 284819 omega = 3; Omega = 3; tau = 8 sigma = 39 918123 659008 838880; phi = 39 852571 442073 216768 ======================================================================= ------ n = 29 greatest prime factor = 29 ------ GPP = 2*3*2*5*3*7*2*3*5*11*3*13*7*5*2*17*3*19*5*7*11*23*3*5*13*3*7*29 = 1156 675078 903494 150000 omega = 10; Omega = 28; tau = 172800 sigma = 7018 084827 412130 304000; phi = 182 693616 744038 400000 ....................................................................... GPP - 1 = 1156 675078 903494 149999 = 443851 * 2605 998587 146349 omega = 2; Omega = 2; tau = 4 sigma = 1156 677684 902081 740200; phi = 1156 672472 904906 559800 ....................................................................... GPP + 1 = 1156 675078 903494 150001 is prime omega = 1; Omega = 1; tau = 2 sigma = 1156 675078 903494 150002; phi = 1156 675078 903494 150000 ======================================================================= ------ page 21 n = 30 greatest prime factor = 5 ------ GPP = 2*3*2*5*3*7*2*3*5*11*3*13*7*5*2*17*3*19*5*7*11*23*3*5*13*3*7*29*5 = 5783 375394 517470 750000 omega = 10; Omega = 29; tau = 201600 sigma = 35092 220881 768130 304000; phi = 913 468083 720192 000000 ....................................................................... GPP - 1 = 5783 375394 517470 749999 = 31 * 37 * 93911 * 299993 * 178 974179 omega = 5; Omega = 5; tau = 32 sigma = 6131 371270 621958 000640; phi = 5445 473461 639436 332800 ....................................................................... GPP + 1 = 5783 375394 517470 750001 = 47 * 928 000331 * 132597 517693 omega = 3; Omega = 3; tau = 8 sigma = 5906 425941 235577 971584; phi = 5660 324848 066414 564560 ======================================================================= ------ page 22 n = 31 greatest prime factor = 5 ------ GPP = 2*3*2*5*3*7*2*3*5*11*3*13*7*5*2*17 _ *3*19*5*7*11*23*3*5*13*3*7*29*5*31 = 179284 637230 041593 250000 omega = 11; Omega = 30; tau = 403200 sigma = 1 122951 068216 580169 728000 phi = 27404 042511 605760 000000 ....................................................................... GPP - 1 = 179284 637230 041593 249999 = 2141 * 806317 * 6 614189 * 15 701603 omega = 4; Omega = 4; tau = 16 sigma = 179368 636963 919078 158560 phi = 179200 637739 957061 818240 ....................................................................... GPP + 1 = 179284 637230 041593 250001 = 2939 * 11839 * 109229 * 47172 675889 omega = 4; Omega = 4; tau = 16 sigma = 179362 429929 091221 120000 phi = 179206 856230 479301 276416 ======================================================================= ------ page 23 n = 32 greatest prime factor = 2 ------ GPP = 2*3*2*5*3*7*2*3*5*11*3*13*7*5*2*17*3 _ *19*5*7*11*23*3*5*13*3*7*29*5*31*2 = 358569 274460 083186 500000 omega = 11; Omega = 31; tau = 483840 sigma = 2 282126 364440 146796 544000 phi = 54808 085023 211520 000000 ....................................................................... GPP - 1 = 358569 274460 083186 499999 = 3 082663 * 116318 025830 291273 omega = 2; Omega = 2; tau = 4 sigma = 358569 390778 109019 873936 phi = 358569 158142 057353 126064 ....................................................................... GPP + 1 = 358569 274460 083186 500001 = 251 * 3209 * 33589 * 1 187623 * 11 159737 omega = 5; Omega = 5; tau = 32 sigma = 360121 077681 600220 473600 phi = 357018 456190 794072 192000 ======================================================================= ------ page 24 n = 33 greatest prime factor = 11 ------ GPP = 2*3*2*5*3*7*2*3*5*11*3*13*7*5*2*17*3*19 _ *5*7*11*23*3*5*13*3*7*29*5*31*2*11 = 3 944262 019060 915051 500000 omega = 11; Omega = 32; tau = 645120 sigma = 25 120548 853687 029399 552000 phi = 602888 935255 326720 000000 ....................................................................... GPP - 1 = 3 944262 019060 915051 499999 = 8191 * 3815 568677 * 126202 964557 omega = 3; Omega = 3; tau = 8 sigma = 3 944743 556204 535106 142208 phi = 3 943780 481917 555033 940640 ....................................................................... GPP + 1 = 3 944262 019060 915051 500001 = 857 * 3847 * 58789 * 20 350106 034371 omega = 4; Omega = 4; tau = 16 sigma = 3 949958 091660 685600 129920 phi = 3 938568 530639 709943 186560 ======================================================================= ------ page 25 n = 34 greatest prime factor = 17 ------ GPP = 2*3*2*5*3*7*2*3*5*11*3*13*7*5*2*17*3*19 _ *5*7*11*23*3*5*13*3*7*29*5*31*2*11*17 = 67 052454 324035 555875 500000 omega = 11; Omega = 33; tau = 967680 sigma = 428 444916 560106 556981 248000 phi = 10 249111 899340 554240 000000 ....................................................................... GPP - 1 = 67 052454 324035 555875 499999 = 37 230797 * 1 800994 330689 067867 omega = 2; Omega = 2; tau = 4 sigma = 67 052456 125029 886601 798664 phi = 67 052452 523041 225149 201336 ....................................................................... GPP + 1 = 67 052454 324035 555875 500001 = 41 * 439 * 631 * 1 930573 * 3 058093 183373 omega = 5; Omega = 5; tau = 32 sigma = 68 953483 732060 000166 607360 phi = 65 164545 282449 989338 278400 ======================================================================= ------ page 26 n = 35 greatest prime factor = 7 ------ GPP = 2*3*2*5*3*7*2*3*5*11*3*13*7*5*2*17*3*19 _ *5*7*11*23*3*5*13*3*7*29*5*31*2*11*17*7 = 469 367180 268248 891128 500000 omega = 11; Omega = 34; tau = 1 161216 sigma = 2999 267377 333298 596675 584000 phi = 71 743783 295383 879680 000000 ....................................................................... GPP - 1 = 469 367180 268248 891128 499999 = 1697 * 557663 * 495974 154719 183009 omega = 3; Omega = 3; tau = 8 sigma = 469 644608 867406 769002 510720 phi = 469 089752 661039 322693 974016 ....................................................................... GPP + 1 = 469 367180 268248 891128 500001 = 547 * 38639 * 1 191373 * 18 640251 216089 omega = 4; Omega = 4; tau = 16 sigma = 470 237819 959851 163158 355200 phi = 468 496586 452504 504949 728128 ======================================================================= ------ page 27 n = 36 greatest prime factor = 3 ------ GPP = 2*3*2*5*3*7*2*3*5*11*3*13*7*5*2*17*3*19*5 _ *7*11*23*3*5*13*3*7*29*5*31*2*11*17*7*3 = 1408 101540 804746 673385 500000 omega = 11; Omega = 35; tau = 1 306368 sigma = 8998 716542 785668 137159 884800 phi = 215 231349 886151 639040 000000 ....................................................................... GPP - 1 = 1408 101540 804746 673385 499999 is prime omega = 1; Omega = 1; tau = 2 sigma = 1408 101540 804746 673385 500000 phi = 1408 101540 804746 673385 499999 ....................................................................... GPP + 1 = 1408 101540 804746 673385 500001 = 13127 * 174 466217 * 614 832964 379839 omega = 3; Omega = 3; tau = 8 sigma = 1408 208816 457658 366417 935360 phi = 1407 994265 153064 646630 783008 ======================================================================= ------ page 28 n = 37 greatest prime factor = 37 ------ GPP = 2*3*2*5*3*7*2*3*5*11*3*13*7*5*2*17*3*19*5*7 _ *11*23*3*5*13*3*7*29*5*31*2*11*17*7*3*37 = 52099 757009 775626 915263 500000 omega = 12; Omega = 36; tau = 2 612736 sigma = 341951 228625 855389 212075 622400 phi = 7748 328595 901459 005440 000000 ....................................................................... GPP - 1 = 52099 757009 775626 915263 499999 = 10613 * 17 923603 * 273887 503640 652241 omega = 3; Omega = 3; tau = 8 sigma = 52104 668967 699506 852816 263152 phi = 52094 845052 399521 985027 909760 ....................................................................... GPP + 1 = 52099 757009 775626 915263 500001 = 47 * 1 454597 * 762 070503 578034 634339 omega = 3; Omega = 3; tau = 8 sigma = 53208 299057 452897 106001 375360 phi = 50991 216486 239363 880597 802608 ======================================================================= ------ page 29 n = 38 greatest prime factor = 19 ------ GPP = 2*3*2*5*3*7*2*3*5*11*3*13*7*5*2*17*3*19*5*7 _ *11*23*3*5*13*3*7*29*5*31*2*11*17*7*3*37*19 = 989895 383185 736911 390006 500000 omega = 12; Omega = 37; tau = 3 919104 sigma = 6 514170 905322 545164 490040 606720 phi = 147218 243322 127721 103360 000000 ....................................................................... GPP - 1 = 989895 383185 736911 390006 499999 = 73 * 1 662257 * 2 125001 * 3838 920596 528159 omega = 4; Omega = 4; tau = 16 sigma = 1 003456 669799 315793 303795 421440 phi = 976334 125650 684000 080544 000000 ....................................................................... GPP + 1 = 989895 383185 736911 390006 500001 = 48563 * 31 546637 * 327 949957 * 1970 258203 omega = 4; Omega = 4; tau = 16 sigma = 989915 801822 345099 371440 383424 phi = 989874 964550 566243 351390 440384 ======================================================================= ------ page 30 n = 39 greatest prime factor = 13 ------ GPP = 2*3*2*5*3*7*2*3*5*11*3*13*7*5*2*17*3*19*5*7*11 _ *23*3*5*13*3*7*29*5*31*2*11*17*7*3*37*19*13 = 12 868639 981414 579848 070084 500000 omega = 12; Omega = 38; tau = 5 225472 sigma = 84 719818 331517 254051 837686 579200 phi = 191383 716318 766037 4343680 000000 ....................................................................... GPP - 1 = 12 868639 981414 579848 070084 499999 = 59 * 1811 * 120 437626 757523 045120 404351 omega = 3; Omega = 3; tau = 8 sigma = 13 093978 781077 905465 490361 149440 phi = 12 643542 057004 769276 740048 663000 ....................................................................... GPP + 1 = 12 868639 981414 579848 070084 500001 = 281 * 673 * 2383 * 5683 * 245131 * 1 250701 * 16 389203 omega = 7; Omega = 7; tau = 128 sigma = 12 941393 393270 030236 915827 253248 phi = 12 796102 611627 189912 884090 880000 ======================================================================= ------ page 31 n = 40 greatest prime factor = 5 ------ GPP = 2*3*2*5*3*7*2*3*5*11*3*13*7*5*2*17*3*19*5*7*11 _ *23*3*5*13*3*7*29*5*31*2*11*17*7*3*37*19*13*5 = 64 343199 907072 899240 350422 500000 omega = 12; Omega = 39; tau = 5 971968 sigma = 423 603429 367807 534774 781686 579200 phi = 9 569185 815938 301871 718400 000000 ....................................................................... GPP - 1 = 64 343199 907072 899240 350422 499999 = 107 * 438287 * 13 241821 * 95 622781 * 1083 555611 omega = 5; Omega = 5; tau = 32 sigma = 64 944692 045694 800027 895177 556992 phi = 63 741710 617027 003332 382040 496000 ....................................................................... GPP + 1 = 64 343199 907072 899240 350422 500001 = 1336 484560 722851 * 48143 616318 524651 omega = 2; Omega = 2; tau = 4 sigma = 64 343199 907072 948720 451301 747504 phi = 64 343199 907072 849760 249543 252500 ======================================================================= ------ page 32 n = 41 greatest prime factor = 41 ------ GPP = 2*3*2*5*3*7*2*3*5*11*3*13*7*5*2*17*3*19*5*7*11*23 _ *3*5*13*3*7*29*5*31*2*11*17*7*3*37*19*13*5*41 = 2638 071196 189988 868854 367322 500000 omega = 13; Omega = 40; tau = 11 943936 sigma = 17791 344033 447916 460540 830836 326400 phi = 382 767432 637532 074868 736000 000000 ....................................................................... GPP - 1 = 2638 071196 189988 868854 367322 499999 = 79 * 331 * 100886 121694 519441 235013 473651 omega = 3; Omega = 3; tau = 8 sigma = 2679 535392 206436 359201 957860 197120 phi = 2596 808772 416930 417389 246811 751000 ....................................................................... GPP + 1 = 2638 071196 189988 868854 367322 500001 = 479 * 2939 * 99901 * 3414 818327 * 5 493055 329623 omega = 5; Omega = 5; tau = 32 sigma = 2644 504605 835442 250993 391019 212800 phi = 2631 641662 619690 333722 484911 099200 ======================================================================= ------ page 33 n = 42 greatest prime factor = 7 ------ GPP = 2*3*2*5*3*7*2*3*5*11*3*13*7*5*2*17*3*19*5*7*11*23 _ *3*5*13*3*7*29*5*31*2*11*17*7*3*37*19*13*5*41*7 = 18466 498373 329922 081980 571257 500000 omega = 13; Omega = 41; tau = 13 934592 sigma = 124540 315585 422310 772360 914835 865600 phi = 2679 372028 462724 524081 152000 000000 ....................................................................... GPP - 1 = 18466 498373 329922 081980 571257 499999 = 617 * 241691 * 537132 458431 * 230 545957 909307 omega = 4; Omega = 4; tau = 16 sigma = 18496 504397 534102 016661 085330 036736 phi = 18436 492596 793288 440174 814825 963200 ....................................................................... GPP + 1 = 18466 498373 329922 081980 571257 500001 = 373 * 224 551559 * 1254 352087 * 175 768152 941789 omega = 4; Omega = 4; tau = 16 sigma = 18516 006509 083540 741391 996359 468800 phi = 18416 990238 096192 441499 319609 458368 ======================================================================= ------ page 34 n = 43 greatest prime factor = 43 ------ GPP = 2*3*2*5*3*7*2*3*5*11*3*13*7*5*2*17*3*19*5*7*11*23*3 _ *5*13*3*7*29*5*31*2*11*17*7*3*37*19*13*5*41*7*43 = 794059 430053 186649 525164 564072 500000 omega = 14; Omega = 42; tau = 27 869184 sigma = 5 479773 885758 581673 983880 252778 086400 phi = 112533 625195 434430 011408 384000 000000 ....................................................................... GPP - 1 = 794059 430053 186649 525164 564072 499999 = 163 * 6317 * 25175 812273 * 30 631695 406547 152153 omega = 4; Omega = 4; tau = 16 sigma = 799057 433492 008880 872521 892419 261792 phi = 789062 968970 387722 845637 818738 306048 ....................................................................... GPP + 1 = 794059 430053 186649 525164 564072 500001 = 2179 * 9109 * 217 250953 * 184 146464 530406 588647 omega = 4; Omega = 4; tau = 16 sigma = 794511 061386 846757 159760 171299 081600 phi = 793607 878735 673654 096870 042767 886208 ======================================================================= ------ page 35 n = 44 greatest prime factor = 11 ------ GPP = 2*3*2*5*3*7*2*3*5*11*3*13*7*5*2*17*3*19*5*7*11*23*3 _ *5*13*3*7*29*5*31*2*11*17*7*3*37*19*13*5*41*7*43*11 = 8 734653 730585 053144 776810 204797 500000 omega = 14; Omega = 43; tau = 34 836480 sigma = 60 281255 758293 687062 507097 999310 848000 phi = 1 237869 877149 778730 125492 224000 000000 ....................................................................... GPP - 1 = 8 734653 730585 053144 776810 204797 499999 = 173 * 331 * 6337 * 15979 234423 * 1 506370 911456 788023 omega = 5; Omega = 5; tau = 32 sigma = 8 813074 790294 236664 295303 251488 502784 phi = 8 656562 005572 627327 478015 007307 786240 ....................................................................... GPP + 1 = 8 734653 730585 053144 776810 204797 500001 = 577670 972464 621571 * 15 120465 017168 560331 omega = 2; Omega = 2; tau = 4 sigma = 8 734653 730585 053160 474946 194430 681904 phi = 8 734653 730585 053129 078674 215164 318100 ======================================================================= ------ page 36 n = 45 greatest prime factor = 5 ------ GPP = 2*3*2*5*3*7*2*3*5*11*3*13*7*5*2*17*3*19*5*7*11*23*3*5 _ *13*3*7*29*5*31*2*11*17*7*3*37*19*13*5*41*7*43*11*5 = 43 673268 652925 265723 884051 023987 500000 omega = 14; Omega = 44; tau = 39 191040 sigma = 301 406896 073107 641236 258174 799310 848000 phi = 6 189349 385748 893650 627461 120000 000000 ....................................................................... GPP - 1 = 43 673268 652925 265723 884051 023987 499999 is prime omega = 1; Omega = 1; tau = 2 sigma = 43 673268 652925 265723 884051 023987 500000 phi = 43 673268 652925 265723 884051 023987 499998 ....................................................................... GPP + 1 = 43 673268 652925 265723 884051 023987 500001 = 17491 * 644 185231 * 88175 279483 * 43 958559 140207 omega = 4; Omega = 4; tau = 16 sigma = 43 675765 620692 922742 360056 491196 960768 phi = 43 670771 685165 417571 936200 315257 528400 ======================================================================= ------ page 37 n = 46 greatest prime factor = 23 ------ GPP = 2*3*2*5*3*7*2*3*5*11*3*13*7*5*2*17*3*19*5*7*11*23*3*5*13 _ *3*7*29*5*31*2*11*17*7*3*37*19*13*5*41*7*43*11*5*23 = 1004 485179 017281 111649 333173 551712 500000 omega = 14; Omega = 45; tau = 58 786560 sigma = 6944 917230 351188 566818 782111 000787 456000 phi = 142 355035 872224 553964 431605 760000 000000 ....................................................................... GPP - 1 = 1004 485179 017281 111649 333173 551712 499999 = 911 * 149940 678426 300959 * 7 353696 210222 871151 omega = 3; Omega = 3; tau = 8 sigma = 1005 587797 215982 854070 750271 953162 199040 phi = 1003 382560 818579 369242 923348 927561 147000 ....................................................................... GPP + 1 = 1004 485179 017281 111649 333173 551712 500001 = 1399 * 86900 747243 417311 * 8 262325 643052 794809 omega = 3; Omega = 3; tau = 8 sigma = 1005 203181 289630 859658 228199 432851 008000 phi = 1003 767176 744931 363657 136600 451166 419040 ======================================================================= ------ page 38 n = 47 greatest prime factor = 47 ------ GPP = 2*3*2*5*3*7*2*3*5*11*3*13*7*5*2*17*3*19*5*7*11*23*3*5*13 _ *3*7*29*5*31*2*11*17*7*3*37*19*13*5*41*7*43*11*5*23*47 = 47210 803413 812212 247518 659156 930487 500000 omega = 15; Omega = 46; tau = 117 573120 sigma = 333356 027056 857051 207301 541328 037797 888000 phi = 6548 331650 122329 482363 853864 960000 000000 ....................................................................... GPP - 1 = 47210 803413 812212 247518 659156 930487 499999 = 71 * 10453 * 21 223613 * 2997 248655 664877 361062 249321 omega = 4; Omega = 4; tau = 16 sigma = 47880 326659 035940 378412 249186 573400 535104 phi = 46541 407456 565639 024289 015116 269511 097600 ....................................................................... GPP + 1 = 47210 803413 812212 247518 659156 930487 500001 = 97 * 8429 * 41 327918 500867 * 1397 172633 948703 437431 omega = 4; Omega = 4; tau = 16 sigma = 47703 171466 559074 726475 396200 103342 912640 phi = 46718 550845 538868 540545 413834 201433 085440 ======================================================================= ------ page 39 n = 48 greatest prime factor = 3 ------ GPP = 2*3*2*5*3*7*2*3*5*11*3*13*7*5*2*17*3*19*5*7*11*23*3*5*13 _ *3*7*29*5*31*2*11*17*7*3*37*19*13*5*41*7*43*11*5*23*47*3 = 141632 410241 436636 742555 977470 791462 500000 omega = 15; Omega = 47; tau = 130 636800 sigma = 1 000101 955373 097000 289032 690394 166034 432000 phi = 19644 994950 366988 447091 561594 880000 000000 ....................................................................... GPP - 1 = 141632 410241 436636 742555 977470 791462 499999 = 8866 109936 826497 * 15 974583 131791 395791 675167 omega = 2; Omega = 2; tau = 4 sigma = 141632 410241 436652 717139 118128 297191 001664 phi = 141632 410241 436620 767972 836813 285733 998336 ....................................................................... GPP + 1 = 141632 410241 436636 742555 977470 791462 500001 = 22 893547 * 6186 564722 427531 074261 056072 734883 omega = 2; Omega = 2; tau = 4 sigma = 141632 416428 001359 170087 051731 847558 128432 phi = 141632 404054 871914 315024 903209 735366 871572 ======================================================================= ------ page 40 n = 49 greatest prime factor = 7 ------ GPP = 2*3*2*5*3*7*2*3*5*11*3*13*7*5*2*17*3*19*5*7*11*23*3*5*13*3 _ *7*29*5*31*2*11*17*7*3*37*19*13*5*41*7*43*11*5*23*47*3*7 = 991426 871690 056457 197891 842295 540237 500000 omega = 15; Omega = 48; tau = 149 299200 sigma = 7 000720 973957 405435 625888 726481 816780 800000 phi = 137514 964652 568919 129640 931164 160000 000000 ....................................................................... GPP - 1 = 991426 871690 056457 197891 842295 540237 499999 = 164 457277 565089 * 6028 476734 923870 638673 512191 omega = 2; Omega = 2; tau = 4 sigma = 991426 871690 062485 674626 766330 636188 577280 phi = 991426 871690 050428 721156 918260 444286 422720 ....................................................................... GPP + 1 = 991426 871690 056457 197891 842295 540237 500001 = 1 069973 * 15 893029 * 58301 696363 901872 032767 478553 omega = 3; Omega = 3; tau = 8 sigma = 991427 860661 906783 154438 839763 382363 355880 phi = 991425 882718 322734 634072 648571 763680 527232 ======================================================================= ------ page 41 n = 50 greatest prime factor = 5 ------ GPP = 2*3*2*5*3*7*2*3*5*11*3*13*7*5*2*17*3*19*5*7*11*23*3*5*13*3 _ *7*29*5*31*2*11*17*7*3*37*19*13*5*41*7*43*11*5*23*47*3*7*5 = 4 957134 358450 282285 989459 211477 701187 500000 omega = 15; Omega = 49; tau = 165 888000 sigma = 35 003619 207270 922634 650249 532881 816780 800000 phi = 687574 823262 844595 648204 655820 800000 000000 ....................................................................... GPP - 1 = 4 957134 358450 282285 989459 211477 701187 499999 = 5 428079 736283 * 913239 045719 102096 117578 719053 omega = 2; Omega = 2; tau = 4 sigma = 4 957134 358451 195525 035178 313579 246845 955336 phi = 4 957134 358449 369046 943740 109376 155529 044664 ....................................................................... GPP + 1 = 4 957134 358450 282285 989459 211477 701187 500001 = 61 * 127 * 3319 * 24967 * 30671 _ * 12 548479 * 20 063398 368463 148219 omega = 7; Omega = 7; tau = 128 sigma = 5 079970 768296 082529 581556 144876 449431 552000 phi = 4 835667 924727 730927 419264 981686 653634 950400 ======================================================================= Appendix: Primes in factorizations of A104350(n)-1 and A104350(n)+1 for n <= 50 --------------------------------------------------------------------- 2, 3, 5, 7, 11 (2x), 13 (2x), 23, 31 (2x), 37 (2x), 41 (2x), 43 (2x), 47 (3x), 53, 59 (2x), 61 (3x), 71 (2x), 73, 79 (2x), 97 (3x), 103 (2x), 107, 127 (2x), 149, 163, 167, 173, 179 (2x), 181 (2x), 191, 211, 223, 227, 229, 251 (2x), 269, 281, 331 (3x), 349, 367, 373, 401, 421, 439, 479, 547, 617, 631, 673, 857, 911, 1217, 1259, 1399, 1543, 1697, 1811, 2129, 2141, 2179, 2243, 2383, 2521, 2789, 2939 (2x), 3209, 3319, 3847, 5683, 5801, 5981, 6317, 6337, 7559, 7561, 8191, 8429, 9067, 9109, 10139, 10453, 10613, 11717, 11839, 13127, 13421, 17491, 17569, 18979, 19069, 24967, 30671, 33589, 37799 (2x), 38639, 40357, 48563, 50261, 58789, 62597, 71437, 76103, 93911, 97103, 99901, 109229, 124759, 241691, 245131, 277021, 299993, 380453, 415147, 415799, 415801, 438287, 443851, 445771, 557663, 594311, 695771, 721381, 806317, 1069973, 1187623, 1191373, 1247401, 1250701, 1389833, 1434493, 1454597, 1646947, 1662257, 1930573, 2125001, 2689891, 3082663, 4939567, 6614189, 6784123, 8937119, 11159737, 12548479, 13241821, 15391787, 15701603, 15893029, 16389203, 17923603, 21223613, 22893547, 31546637, 37230797, 46313123, 95622781, 174466217, 178974179, 217250953, 224551559, 327949957, 644185231, 928000331, 1083555611, 1135133999, 1254352087, 1970258203, 2129284819, 2217247573, 3414818327, 3815568677, 5464930609, 15979234423, 16347569521, 25175812273, 47172675889, 88175279483, 126202964557, 132597517693, 537132458431, 938977307561, 1099944846001, 1687166397251, 3058093183373, 5428079736283, 5493055329623, 5499724229999, 6121943187511, 10603380543703, 13027211250107, 18640251216089, 20350106034371, 41327918500867, 43958559140207, 164457277565089, 175768152941789, 230545957909307, 319699160368361, 614832964379839, 1336484560722851, 2605998587146349, 3838920596528159, 8866109936826497, 29220034833989999, 48143616318524651, 86900747243417311, 116318025830291273, 146100174169950001, 149940678426300959, 273887503640652241, 495974154719183009, 577670972464621571, 1506370911456788023, 1800994330689067867, 7353696210222871151, 8262325643052794809, 15120465017168560331, 20063398368463148219, 30631695406547152153, 184146464530406588647, 762070503578034634339, 1156675078903494150001, 1397172633948703437431, 15974583131791395791675167, 120437626757523045120404351, 1408101540804746673385499999, 2997248655664877361062249321, 6028476734923870638673512191, 58301696363901872032767478553, 100886121694519441235013473651, 913239045719102096117578719053, 6186564722427531074261056072734883, 43673268652925265723884051023987499999. Example: A000040(3998)=37799 occurs two times 1. A104350(10)-1 = 37799 [page 11] 2. A104350(27)-1 = 401*37799*76103*4939567 [page 19]