A395982 Primes p such that the Fermat quotient q = (2^(p-1) - 1)/p mod p satisfies 1 < q < p and q divides p - 1.

7, 11, 13, 19, 31, 71, 127, 379, 491, 2633, 2659, 8191, 13249, 26893, 70687, 74597, 87211, 131071, 184511, 524287, 642581, 1897121, 2676301, 2703739, 15456151, 52368101, 102785339, 126233057, 193481677, 856645921, 1552107133, 2147483647, 2935442621, 3668158729

Offset

1,1

Comments

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

Links

Table of n, a(n) for n=1..34.

Formula

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

Mathematica

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

Prog

(Python)
from sympy import primerange
def sequence(limit):
    results = []
    for p in primerange(3, limit):
        q = ((pow(2, p - 1, p**2) - 1) // p) % p
        if q > 1 and (p - 1) % q == 0:
            results.append(p)
    return results
print(sequence(3000000))
(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

Crossrefs

Cf. A000668, A001220, A007663, A130912.

Sequence in context: A059308 A075521 A084444 * A081091 A027901 A129213

Adjacent sequences: A395979 A395980 A395981 * A395983 A395984 A395985

Keyword

nonn,changed

Author

Vincenzo Manto, May 13 2026

Extensions

a(25)-a(27) from Amiram Eldar, May 21 2026

a(28)-a(29) from Vincenzo Manto, May 22 2026

a(30)-a(34) from Jinyuan Wang, Jul 10 2026

Status

approved