p!/p#+1 - INTEGERS

素数 $p$ に対して、 $a_p$ を

$\displaystyle a_p:=\frac{p!}{p\#}+1$

で定義します。 $p\#$ は素数階乗です。

$a_2=2$

$a_3=2$

$a_5=5$

$a_7=25=5^2$

$a_{11}=17281=11\times 1571$

$a_{13}=207361 = 7 \times 11 \times 2693$

$a_{17}=696 729601 = 19 \times 2377 \times 15427$

$a_{19}=12541 132801 = 31 \times 61 \times 239 \times 27749$

$a_{23}=115 880067 072001 = 27893 \times 4154 449757$

$a_{29}=1366 643159 020339 200001 = 223 \times 40429 \times 151 585363 459003$

$a_{31}=40999 294770 610176 000001 = 554797 \times 73899 633146 196133$

$\begin{align}a_{37}&=1 854768 736099 424576 471040 000001 \\ &= 10333 \times 17551 \times 412 876421 \times 24 770877 398207\end{align}$

$\begin{align}a_{41}&=109950 690675 973888 893203 251200 000001 \\ &= 29 \times 743 \times 3 570470 221379 \times 1 429176 066859 677377\end{align}$

$\begin{align}a_{43}&=4 617929 008390 903333 514536 550400 000001 \\ &= 61 \times 613 \times 1 100653 \times 14 481017 \times 7 748318 379505 746557\end{align}$

$\begin{align}a_{47}&=420600 974084 243475 616503 989010 432000 000001 \\ &= 28 652131 \times 677643 511121 \times 21662 673895 172922 098651\end{align}$

$\begin{align}a_{53}&=131 175012 912718 250806 592304 873426 282086 400000 000001 \\ &= 293 049347 \times 447620 901583 917369 406703 727865 417431 168683\end{align}$

$\begin{align}a_{59}&=72127 095056 108748 061547 940676 488545 036232 818688 000000 000001\\ &= 7 731517 \times 347785 835587 \\ &\quad \times 26823 894708 473887 265117 477378 744683 922119\end{align}$

$\begin{align}a_{61}&=4 327625 703366 524883 692876 440589 312702 173969 121280 000000 000001 \\ & = 1483 \times 7127 \times 7 008809 876989 \\ &\quad \times 58419 455175 821973 834615 220346 798867 768249\end{align}$

$\begin{align}a_{67}&=4641 081518 632318 859492 593303 493155 822657 704661 332315 340800 000\\ &\quad \ 000 000001 \\ &= 73 \times 463 \times 292069 \times 1 485559 \times 1684 513433 610479 \\ &\quad \times 187 873487 957710 251112 719111 051932 885411\end{align}$

$\begin{align}a_{71}&=1524 316813 979598 806211 747344 599292 098393 696518 967985 650532 352\\ &\quad \ 000 000000 000001 \\ &= 337809 273983 \times 5 530881 877187 \times 31 734470 040281 \times 10638 349187 975557 \\ &\quad \times 2416 594580 545944 177593\end{align}$

$\begin{align}a_{73}&=109750 810606 531114 047245 808811 149031 084346 149365 694966 838329 3\\ &\quad \ 44000 000000 000001\\ &= 67 \times 1861 \times 8539 \times 678253 \\ &\quad \times 151 980484 198948 823551 696833 505973 745524 167842 228692 010030 901969\end{align}$

$\begin{align}a_{79}&=278 035108 834491 952374 200462 632195 223836 131498 049515 522896 5183\\ &\quad \ 47 698995 200000 000000 000001\\ & = 440939 \times 229 918267 \times 1 800121 214093 \times 3745 828933 595186 733991 167749 \\ &\quad \times 406 722276 544872 969063 670718 259161\end{align}$

$\begin{align}a_{83}&=147 736735 430295 643813 555157 824243 254137 566832 803590 568246 2939\\ &\quad \ 89 233338 089472 000000 000000 000001 \\ &= 1321 \times 3011 \times 19433 \times 11078 093307 398182 946017 \\ &\quad \times 172532 158443 057699 772164 532525 607875 550686 109683 184687 080411\end{align}$

$\begin{align}a_{89}&=694523 725020 825051 446420 532149 209597 411955 055040 671457 338706 5\\ &\quad \ 88955 510775 037065 953280 000000 000000 000001 \\ &= 94099 \times 6 218647 \times 18 494666 981437 \times 58 443131 174093 \\ &\quad \times 1489 840216 743290 665508 300119 \\ &\quad \times 737032 243754 315571 280907 352212 738723\end{align}$

$\begin{align}a_{97} &= 41 721936 280667 021214 842775 197740 704647 938666 856859 336935 44924\\ &\quad \ 8 035545 074425 279071 887258 996965 376000 000000 000000 000001 \\ &= 89 \times 647 \times 6449 \times 196 829639 \\ &\quad \times 570 804377 775579 113845 140084 714923 228047 667959 084711 538825 24\\ &\quad \ 2983 631969 831787 242913 416490 381542 591177\end{align}$