A395982 - OEIS

%I #34 Jul 10 2026 10:24:29

%S 7,11,13,19,31,71,127,379,491,2633,2659,8191,13249,26893,70687,74597,

%T 87211,131071,184511,524287,642581,1897121,2676301,2703739,15456151,

%U 52368101,102785339,126233057,193481677,856645921,1552107133,2147483647,2935442621,3668158729

%N Primes p such that the Fermat quotient q = (2^(p-1) - 1)/p mod p satisfies 1 < q < p and q divides p - 1.

%C All known Mersenne primes (A000668) > 3 are terms of this sequence.

%F { p prime : q = (2^(p-1) - 1)/p mod p, 1 < q < p and (p-1) mod q = 0 }.

%t Select[Prime[Range[10000]], 1 < (q = Mod[(2^(# - 1) - 1)/#, #]) && Divisible[# - 1, q] &] (* _Amiram Eldar_, May 21 2026 *)

%o (Python)

%o from sympy import primerange

%o def sequence(limit):

%o     results = []

%o     for p in primerange(3, limit):

%o         q = ((pow(2, p - 1, p**2) - 1) // p) % p

%o         if q > 1 and (p - 1) % q == 0:

%o             results.append(p)

%o     return results

%o print(sequence(3000000))

%o (PARI) isok(p) = if (isprime(p) && (p>2), my(q=((2^(p-1)-1)/p) % p); (q>1) && (q<p) && !((p-1) % q)); \\ _Michel Marcus_, May 13 2026

%Y Cf. A000668, A001220, A007663, A130912.

%K nonn,changed

%O 1,1

%A _Vincenzo Manto_, May 13 2026

%E a(25)-a(27) from _Amiram Eldar_, May 21 2026

%E a(28)-a(29) from _Vincenzo Manto_, May 22 2026

%E a(30)-a(34) from _Jinyuan Wang_, Jul 10 2026