Vincenzo Manto

A395982

Created: 5/27/2026 | Author: Vincenzo Manto , May 13 2026

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

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

Sequence Chart

Graph of A395982

Data

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

Formula

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

Computational Implementations

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))

CODE

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

Mathematica

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

Cross-References

See also OEIS entries: Cf. A000668, A007663, A001220, A000040, A130912.