A089162 -id:A089162 - cp's OEIS frontend

A122094 Prime divisors of Mersenne numbers. Primes p such that the multiplicative order of 2 modulo p is prime.

3, 7, 23, 31, 47, 89, 127, 167, 223, 233, 263, 359, 383, 431, 439, 479, 503, 719, 839, 863, 887, 983, 1103, 1319, 1367, 1399, 1433, 1439, 1487, 1823, 1913, 2039, 2063, 2089, 2207, 2351, 2383, 2447, 2687, 2767, 2879, 2903, 2999, 3023, 3119, 3167, 3343

Offset: 1

Max Alekseyev, Oct 25 2006

nonn

Except for the first term (3), all terms are 1 or 7 (mod 8). - William Hu, May 03 2024

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)

Cf. A001348, A016047, A003260, A000668, A137332.

Cf. A089162 (this list sorted by q).

[p: p in PrimesInInterval(2, 4000) | IsPrime(Modorder(2, p))]; // Vincenzo Librandi, Oct 28 2016

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 *)
Select[Prime@ Range@ 500, PrimeQ@ MultiplicativeOrder[2, #] &] (* Michael De Vlieger, Oct 28 2016 *)

forprime(p=3,10^5,if(isprime(znorder(Mod(2,p))),print1(p,",")))

p is a prime divisor of a Mersenne number 2^q - 1 iff prime q is the multiplicative order of 2 modulo p.

A244453 Prime factors of 2^A054723(n)-1, ordered by increasing n, then by increasing size of the factors.

Original entry on oeis.org

23, 89, 47, 178481, 233, 1103, 2089, 223, 616318177, 13367, 164511353, 431, 9719, 2099863, 2351, 4513, 13264529, 6361, 69431, 20394401, 179951, 3203431780337, 193707721, 761838257287, 228479, 48544121, 212885833

Offset: 1

Felix Fröhlich, Jun 28 2014

Subsequence of A060443.

Prime factors of composite Mersenne numbers; A089162 with the Mersenne primes A000668 removed. - Jens Kruse Andersen, Jul 11 2014

			A054723(1) = 11. 2^11-1 = 2047 = 23*89. - _Jens Kruse Andersen_, Jul 11 2014
Triangle begins:
23, 89;
47, 178481;
233, 1103, 2089;
223, 616318177;
13367, 164511353;
431, 9719, 2099863;
2351, 4513, 13264529;
6361, 69431, 20394401;

Jens Kruse Andersen, Table of n, a(n) for n = 1..577
Sergey Nikitin, Euler-Fermat algorithm and some of its applications, 2018.
Sam Wagstaff, The Cunningham Project

Cf. A003260, A016047, A046800, A089162.

Map[FactorInteger, Select[2^Prime@Range@20 - 1, CompositeQ]][[All, All, 1]] // Flatten (* Michael De Vlieger, Nov 20 2018 *)

forprime(n=1, 100, m=2^n-1; if(!isprime(m), f=factor(m); for(i=1, #f~, print1(f[i,1]", ")))) \\ Jens Kruse Andersen, Jul 11 2014

cp's OEIS Frontend

A122094 Prime divisors of Mersenne numbers. Primes p such that the multiplicative order of 2 modulo p is prime.

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