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A186283 Least number k such that k*n+1 is a prime dividing 2^n-1.

%I A186283 #43 Feb 16 2025 08:33:14
%S A186283 1,2,1,6,1,18,2,8,1,2,1,630,3,2,1,7710,1,27594,2,6,1,2,10,24,105,9728,
%T A186283 1,8,1,69273666,8,18166,1285,2,1,6,4599,2,1,326,1,10,2,14,1,50,2,
%U A186283 90462791808,5,2,1,120,1615,16,2,568,1,3050,1,37800705069076950,11545611,2,4,126,1,2891160,2,145690999102,1
%N A186283 Least number k such that k*n+1 is a prime dividing 2^n-1.
%C A186283 The smallest prime factor of 2^n-1 of the form k*n+1 is A186522(n).
%C A186283 By Fermat's little theorem, a(n) = 1 if and only if n+1 is an odd prime. Further, for prime p, a(p) = 2 if and only if p is in A002515. - _Thomas Ordowski_, Sep 03 2017
%D A186283 Kenneth H. Rosen, Elementary Number Theory and Its Applications, 3rd Ed, Theorem 6.12, p. 225
%H A186283 Max Alekseyev, <a href="/A186283/b186283.txt">Table of n, a(n) for n = 2..1236</a> (terms to a(300) from David A. Corneth)
%H A186283 Will Edgington, <a href="https://web.archive.org/web/20060630100606/http://www.garlic.com:80/~wedgingt/mersenne.html">Mersenne Page</a> [from Internet Archive Wayback Machine]
%H A186283 Eric W. Weisstein, <a href="https://mathworld.wolfram.com/MersenneNumber.html">MathWorld: Mersenne Number</a>
%H A186283 Eric W. Weisstein, <a href="https://mathworld.wolfram.com/MersennePrime.html">MathWorld: Mersenne Prime</a>
%F A186283 a(n) = (A186522(n)-1)/n.
%e A186283 For n=8, 2^n-1 = 255 = 3 * 5 * 17.  The smallest prime factor of the form k*n+1 is 17 = 2*8+1. Hence, a(8) = 2.
%t A186283 Table[p=First/@FactorInteger[2^n-1]; (Select[p, Mod[#1,n] == 1 &, 1][[1]] - 1)/n, {n, 2, 70}]
%o A186283 (PARI) a(n) = {if(isprime(n+1),return(1)); my(f = factor(2^n - 1)[,1]); for(i=1,#f, if(f[i]%n == 1, return((f[i]-1) / n)))} \\ _David A. Corneth_, Sep 03 2017
%Y A186283 Cf. A000225, A049479, A079324.
%K A186283 nonn
%O A186283 2,2
%A A186283 _Bill McEachen_, Feb 16 2011