A365098 Primes p such that Sum_{k=1..p-1} q^2_p(k) == 0 (mod p), with q_p(k) a Fermat quotient.

2, 11, 971

Offset: 1

Author

Felix Fröhlich, Aug 21 2023

Keywords

• nonn
• hard
• more
• bref

Comments

The congruence in the definition is given in Gy, 2018, Eq. 16.

The terms, except for the prime 2, satisfy the congruence B_{p-1} - 1 + 1/p == (B_{2p-2} - 1 + 1/p)/2 (mod p^2), with B_i a Bernoulli number (cf. Gy, 2018, Eq. 18).

Any odd prime that is a term of both A007540 and A197632, i.e., that is simultaneously a Wilson prime and a Lerch prime, is in this sequence (cf. Gy, 2018, Theorem 5).

An equivalent definition, better suited for computational purposes, is: "Primes p such that Sum_{k=1..p-1} (k^(p-1) - 1)^2 == 0 (mod p^3)." - John Blythe Dobson, Apr 30 2024

a(4) > 427000, if it exists (Gy, 2018). - Amiram Eldar, Aug 22 2023

a(4) > 39540000, if it exists. - John Blythe Dobson, Apr 30 2024

Links

• René Gy, Generalized Lerch primes, INTEGERS, 18 (2018), #A10.

Crossrefs

Cf. A007540, A197632.

Programs

Mathematica

Join[{2}, Select[Prime[Range[2, 200]], Divisible[Numerator[BernoulliB[# - 1] - 1 + 1/# - (BernoulliB[2*# - 2] - 1 + 1/#)/2], #^2] &]] (* Amiram Eldar, Aug 22 2023 *)

PARI

forprime(p = 2, 10000, if(sum(j=1, p-1, (Mod(j, p^3)^(p-1) - 1)^2) % p^3 == 0, print1(p, ", "))) /* John Blythe Dobson, Apr 30 2024 */