need more terms for A084598 & A084599

[all at abouthugo.de]

mlb at fxpt.com schrieb am 31.05.2003, 20:33:49:
> I just submitted A084598 & A084599, which unfortunately match for the first
> 15 terms I was able to quickly calculate!  Any help extending these, especially
> to their point of divergence, will be greatly appreciated.  (As the comment
> on the related pair A000945 & A000946 says "The computational problem inherent
> in continuing the sequence further is the enormous size of the numbers that
> must be factored.")  Thanks!
> 
> %I A084599
> %S A084599 2 3 5 29 11 7 13 37 32222189 131 136013303998782209 31 197 19
> 157
> %N A084599 a(1) = 2, a(2)=3, a(n+1) is largest prime factor of Product_{k=1..n}
> a(k) - 1
> %C A084599 Like the Euclid-Mullin sequence A000946, but subtracting rather
> than adding 1 to the product.  When does this sequence diverge from A084598?
> %e A084599 a(4)=29 since 2*3*5=30 and 29 is the smallest prime factor of
> 30-1
> %Y A084599 Cf. A000946, A084598
> %O A084599 1
> %K A084599 ,more,nonn,
> %A A084599 Marc LeBrun (mlb at well.com), May 31 2003

Marc, SeqFans,

I looked up the definitions of A00945 and A00946:

A00945: Euclid-Mullin sequence: a(1) = 2, a(n+1) is smallest prime
factor of
Product_{k=1..n} a(k) + 1.

I guess what you wanted to create was the analoguous
%I A084598
%N a(1) = 2, a(n+1) is _smallest_ prime factor of
Product_{k=1..n} a(k) - 1.

Omitting the first steps (I got the same results as given above)
(What you computed was the smallest factor contradicting the
given description saying _largest_)

Hope the following is below Olivier's length limit. Sorry if line
wrapping becomes ugly, but
I can't control it with my Web-based mail interface.

2*3*5*29*11*7*13*37*32222189*131*136013303998782209*31*197*19*157-1=
337010 905288 619443 223826 542419 088810 172089 = 17 x 452704 788101 x
43790 
504143 967027 283161 477717 

2*3*5*29*11*7*13*37*32222189*131*136013303998782209*31*197*19*157*17-1=
5 729185 389906 530534 805051 221124 509772 925529 = 8609 x 32183 x 8
907623 
x 2321 409806 422010 530425 341209

2*3*5*29*11*7*13*37*32222189*131*136013303998782209*31*197*19*157*17*8609-1=
49322 557021 705321 374136 685962 660904 635115 887769 = 1 831129 x 96
593227 
x 395 499093 031447 x 705073 635630 813269

2*3*5*29*11*7*13*37*32222189*131*136013303998782209*31*197*19*157*17*8609*1831129-1
90315 964516 598243 422501 535630 121299 643595 120456 392329 = 35977 x
30 
902882 521913 x 12326 099580 658421 x 6 590447 658135 399749

2*3*5*29*11*7*13*37*32222189*131*136013303998782209*31*197*19*157*17*8609*1831129*35977-1=
3249 297455 413655 003611 337747 364873 997277 621648 659626 856409 =
508 
326079 288931 x 8 888176 173420 238273 x 719174 739667 579660 597843

2*3*5*29*11*7*13*37*32222189*131*136013303998782209*31*197*19*157*17*8609*1831129*35977*508326079288931-1
=1 651702 635953 923334 134604 910911 106408 049070 496016 084226 090865

963639 397709 = 487 x 4783 x 317419 x 2 233930 207153 358200 864647
101061 
036446 340222 249817 282225 419791

2*3*5*29*11*7*13*37*32222189*131*136013303998782209*31*197*19*157*17*8609*1831129*35977*508326079288931*487-1
804 379183 709560 663723 552591 613708 820719 897331 559833 018106
251724 
292386 684769 = 10253 x 112687 x 24025 694597 x 28 977445 414406 004620 
114205 855735 530722 214246 007446 549807

2*3*5*29*11*7*13*37*32222189*131*136013303998782209*31*197*19*157*17*8609*1831129*35977*508326079288931*487*10253-1
=8 247299 770574 125485 157584 721815 356538 841107 340482 967934 643398

929169 840678 946809 = 1 390043 x 5 933125 644727 627479 982694 579819
010303 
164080 061180 098698 129049 913686 008763 (Composite)

2*3*5*29*11*7*13*37*32222189*131*136013303998782209*31*197*19*157*17*8609*1831129*35977*508326079288931*487*10253*1390043-1
=11 464101 314988 169111 764904 539466 383649 320309 370886 966196
775514 
177700 032846 885260 612829 = 18 122659 735201 507243 x 632583 819510
789865 
322650 757419 458606 291647 205204 259379 895384 247703

2*3*5*29*11*7*13*37*32222189*131*136013303998782209*31*197*19*157*17*8609*1831129*35977*508326079288931*487*10253*1390043*18122659735201507243-1
=207 760007 301406 743812 608490 818835 654125 505586 413871 217895
844194 
645072 449534 280678 373100 955403 477863 727689 = 25 319167 x 5211
496051 x 
1574 527048 014723 655744 894809 010028 234607 779603 735188 723067
993331 
116600 352976 240928 012717 (Composite) 

2*3*5*29*11*7*13*37*32222189*131*136013303998782209*31*197*19*157*17*8609*1831129*35977*508326079288931*487*10253*1390043*18122659735201507243*25319167-1
=5260 310320 785536 681517 651084 660066 672357 914901 845736 482398
267770 
199095 076857 524720 601831 397720 208412 524625 634229 = 9512 386441 x 
552995 860020 226514 420015 990250 292089 910891 258107 344640 667365
815042 
700185 108840 525143 025084 310723 537869 (Composite)

2*3*5*29*11*7*13*37*32222189*131*136013303998782209*31*197*19*157*17*8609*1831129*35977*508326079288931*487*10253*1390043*18122659735201507243*25319167*9512386441-1
=50 038104 570892 699598 176639 479889 361308 293419 211349 229588
824317 
499129 175879 569371 041075 174347 441788 814993 383461 650477 475429 =
85577 
x 584 714404 231191 787491 693322 737293 446934 262935 267060 420309
479386 
974644 774642 361511 166261 663150 633801 313359 704846 518077
(Composite)

2*3*5*29*11*7*13*37*32222189*131*136013303998782209*31*197*19*157*17*8609*1831129*35977*508326079288931*487*10253*1390043*18122659735201507243*25319167*9512386441*85577-1
=4 282110 874863 284553 513162 276770 491872 679825 935849 633020 522818

618622 977484 245908 065582 090195 131025 961420 688776 497662 910914
873109 
= 1031 x 1787 x 274 100051 x 8479 408809 157692 145898 879272 416881
417830 
313382 292264 846205 285083 062056 974055 469000 116275 023820 736061
153888 
792547 (Composite)

2*3*5*29*11*7*13*37*32222189*131*136013303998782209*31*197*19*157*17*8609*1831129*35977*508326079288931*487*10253*1390043*18122659735201507243*25319167*9512386441*85577*1031-1
=4414 856311 984046 374672 070307 350377 120732 900539 860971 644159
025995 
800289 786257 531215 615134 991180 087766 224730 128569 090461 153234
176409 
= 3650 460767 x 1 209397 003220 565327 246008 533078 524966 580718
915547 
510017 701561 561749 086943 757182 753366 962837 209445 282671 279323
618899 
877127 (Composite)

2*3*5*29*11*7*13*37*32222189*131*136013303998782209*31*197*19*157*17*8609*1831129*35977*508326079288931*487*10253*1390043*18122659735201507243*25319167*9512386441*85577*1031*3650460767-1
16 116259 758840 073220 648975 147648 183401 889875 706875 596621 502009

106602 064631 963933 460880 868056 711801 422220 045039 504330 577313
818936 
148261 906469 = 107 x 150619 250082 617506 735037 150912 599844 877475
473896 
033613 285065 505669 178174 130504 051036 269794 922540 200207 663972
331816 
173619 755317 160264 129967 (Composite)

A084598 becomes therefore:
2 3 5 29 11 7 13 37 32222189 131 136013303998782209 31 197 19 157
17 8609 1831129 35977 508326079288931 487 10253 1390043
18122659735201507243
25319167 9512386441 85577 1031 3650460767 107

A00946: Euclid-Mullin sequence: a(1) = 2, a(n+1) is largest prime factor
of Product_{k=1..n} a(k) + 1.

A084599 with same definition, but replaces a(k)+1 by a(k)-1

 From this point the sequence A084599 starts to differ from A084598:

2*3*5*29-1=869 = 11 x 79
 
2*3*5*29*79-1=68729 is prime

2*3*5*29*79*68729-1=4723 744169 = 61 x 139 x 149 x 3739

2*3*5*29*79*68729*3739-1=17 662079 451629 = 2 839019 x 6 221191

2*3*5*29*79*68729*3739*6221191-1=109 879169 725765 491329
 = 83 x 8423 x 157 170297 801581
2*3*5*29*79*68729*3739*6221191*157170297801581-1=
17269 741827 989025 572635 853315 792729 =
 41 x 5955 703423 x 70724 343608 203457 341903

2*3*5*29*79*68729*3739*6221191*157170297801581*70724343608203457341903-1=
1221 391155 067659 532019 870755 557274 752597 357237 697091 765189 =
 7 x 349 x 449 x 112939 x 9 937441 x 21 420649 x 46 316297 682014 731387
158877 659877

2*3*5*29*79*68729*3739*6221191*157170297801581*70724343608203457341903*46316297682014731387158877659877-1
=56570 316324 293534 522034 875125 819032 625802 887139 428502 972592
854382 
687429 806751 914368 281629 = 7 x 257 x 521 x 682511 x 10829 594203 x 50

852665 316801 x 2043 158415 368893 790939 x 78 592684 042614 093322
289223 662773

2*3*5*29*79*68729*3739*6221191*157170297801581*70724343608203457341903*46316297682014731387158877659877*78592684042614093322289223662773-1
=4 446012 997065 936021 048244 147677 717716 431750 551697 630262 292281

015505 778394 524122 531060 076158 474920 181837 998919 482610 759989 =
7 x 
11 x 17 x 86599 x 294757 x 933418 660159 x 9669 562218 961751 x 2 289336

175732 053683 x 35403 807765 085882 291423 x 181 891012 640244 955605
725966 
274974 474087

Thats what I got for A084599:
2 3 5 29 79 68729 3739 6221191 157170297801581 70724343608203457341903
46316297682014731387158877659877 78592684042614093322289223662773
181891012640244955605725966274974474087

All calculations done with Dario Alpern's ECM:
http://www.alpertron.com.ar/ECM.HTM

Hugo Pfoertner


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