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%I #36 May 09 2024 13:26:41
%S 3,7,23,31,47,89,127,167,223,233,263,359,383,431,439,479,503,719,839,
%T 863,887,983,1103,1319,1367,1399,1433,1439,1487,1823,1913,2039,2063,
%U 2089,2207,2351,2383,2447,2687,2767,2879,2903,2999,3023,3119,3167,3343
%N Prime divisors of Mersenne numbers. Primes p such that the multiplicative order of 2 modulo p is prime.
%C Except for the first term (3), all terms are 1 or 7 (mod 8). - _William Hu_, May 03 2024
%H Charles R Greathouse IV, <a href="/A122094/b122094.txt">Table of n, a(n) for n = 1..10000</a> (first 1000 terms from T. D. Noe)
%F p is a prime divisor of a Mersenne number 2^q - 1 iff prime q is the multiplicative order of 2 modulo p.
%t Reap[For[p=2, p<10^5, p=NextPrime[p], If[PrimeQ[MultiplicativeOrder[2, p]], Sow[p]]]][[2, 1]] (* _Jean-François Alcover_, Dec 10 2015 *)
%t Select[Prime@ Range@ 500, PrimeQ@ MultiplicativeOrder[2, #] &] (* _Michael De Vlieger_, Oct 28 2016 *)
%o (PARI) forprime(p=3,10^5,if(isprime(znorder(Mod(2,p))),print1(p,",")))
%o (Magma) [p: p in PrimesInInterval(2, 4000) | IsPrime(Modorder(2, p))]; // _Vincenzo Librandi_, Oct 28 2016
%Y Cf. A001348, A016047, A003260, A000668, A137332.
%Y Cf. A089162 (this list sorted by q).
%K nonn
%O 1,1
%A _Max Alekseyev_, Oct 25 2006