

A365098


Primes p such that Sum_{k=1..p1} q^2_p(k) == 0 (mod p), with q_p(k) a Fermat quotient.


0




OFFSET

1,1


COMMENTS

The congruence in the definition is given in Gy, 2018, Eq. 16.
The terms, except for the prime 2, satisfy the congruence B_{p1}  1 + 1/p == (B_{2p2}  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..p1} (k^(p1)  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


LINKS



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 *)


PROG

(PARI) forprime(p = 2, 10000, if(sum(j=1, p1, (Mod(j, p^3)^(p1)  1)^2) % p^3 == 0, print1(p, ", "))) /* John Blythe Dobson, Apr 30 2024 */


CROSSREFS



KEYWORD

nonn,hard,more,bref


AUTHOR



STATUS

approved



