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%I #40 May 06 2021 23:10:27
%S 1,1,1,2,1,1,2,1,3,10,3,1,2,1,1,4,1,29,14,1,1,14,2,1,2,4,1,2,4,5,12,2,
%T 1,2,2,9,16,1,2,80,1,2,4,2,3,16,2,2,2,1,15,960,15,1,4,3,1,14,1,6,20,1,
%U 3,946,6,1,18,10,1,4,1,5,42,4,1,828,1,1,2,1,12,2,6,4,30,3,3022,2,1,1
%N Smallest k >= 1 such that (n-1)*n^k + 1 is prime.
%C a(prime(j)) + 1 = A087139(j).
%C a(123) > 10^5, a(342) > 10^5, see the Barnes link for the Sierpinski base-123 and base-342 problems.
%C a(251) > 73000, see A087139.
%H Eric Chen, <a href="/A305531/b305531.txt">Table of n, a(n) for n = 2..122</a>
%H Gary Barnes, <a href="http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm">Sierpinski conjectures and proofs</a>
%H Eric Chen, <a href="/A305531/a305531.txt">Table n, a(n) for n = 2..360 status</a>
%o (PARI) a(n)=for(k=1,2^16,if(ispseudoprime((n-1)*n^k+1),return(k)))
%Y For the numbers k such that these forms are prime:
%Y a1(b): numbers k such that (b-1)*b^k-1 is prime
%Y a2(b): numbers k such that (b-1)*b^k+1 is prime
%Y a3(b): numbers k such that (b+1)*b^k-1 is prime
%Y a4(b): numbers k such that (b+1)*b^k+1 is prime (no such k exists when b == 1 (mod 3))
%Y a5(b): numbers k such that b^k-(b-1) is prime
%Y a6(b): numbers k such that b^k+(b-1) is prime
%Y a7(b): numbers k such that b^k-(b+1) is prime
%Y a8(b): numbers k such that b^k+(b+1) is prime (no such k exists when b == 1 (mod 3)).
%Y Using "-------" if there is currently no OEIS sequence and "xxxxxxx" if no such k exists (this occurs only for a4(b) and a8(b) for b == 1 (mod 3)):
%Y .
%Y b a1(b) a2(b) a3(b) a4(b) a5(b) a6(b) a7(b) a8(b)
%Y --------------------------------------------------------------------
%Y 2 A000043 ------- A002235 A002253 A000043 ------- A050414 A057732
%Y 3 A003307 A003306 A005540 A005537 A014224 A051783 A058959 A058958
%Y 4 A272057 ------- ------- xxxxxxx A059266 A089437 A217348 xxxxxxx
%Y 5 A046865 A204322 A257790 A143279 A059613 A124621 A165701 A089142
%Y 6 A079906 A247260 ------- ------- A059614 A145106 A217352 A217351
%Y 7 A046866 A245241 ------- xxxxxxx A191469 A217130 A217131 xxxxxxx
%Y 8 A268061 A269544 ------- ------- A217380 A217381 A217383 A217382
%Y 9 A268356 A056799 ------- ------- A177093 A217385 A217493 A217492
%Y 10 A056725 A056797 A111391 xxxxxxx A095714 A088275 A092767 xxxxxxx
%Y 11 A046867 A057462 ------- ------- ------- ------- ------- -------
%Y 12 A079907 A251259 ------- ------- ------- A137654 ------- -------
%Y 13 A297348 ------- ------- xxxxxxx ------- ------- ------- xxxxxxx
%Y 14 A273523 ------- ------- ------- ------- ------- ------- -------
%Y 15 ------- ------- ------- ------- ------- ------- ------- -------
%Y 16 ------- ------- ------- xxxxxxx ------- ------- ------- xxxxxxx
%Y Cf. (smallest k such that these forms are prime) A122396 (a1(b)+1 for prime b), A087139 (a2(b)+1 for prime b), A113516 (a5(b)), A076845 (a6(b)), A178250 (a7(b)).
%K nonn
%O 2,4
%A _Eric Chen_, Jun 04 2018
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