cp's OEIS Frontend

A258367 a(n) is the smallest A (in absolute value) such that for p = prime(n), 2^{(p-1)/2} == +-1 + A*p (mod p^2), i.e., such that p is a near-Wieferich prime.

%I A258367 #34 Sep 01 2022 09:33:27
%S A258367 1,1,1,3,5,2,8,3,14,3,18,9,9,22,18,4,18,5,1,28,30,24,3,20,46,22,47,21,
%T A258367 15,9,57,42,15,48,28,41,48,60,85,25,74,25,52,11,32,51,17,13,34,113,13,
%U A258367 71,2,16,64,130,81,35,37,29,39,147,68,60,71,96,92,99,12
%N A258367 a(n) is the smallest A (in absolute value) such that for p = prime(n), 2^{(p-1)/2} == +-1 + A*p (mod p^2), i.e., such that p is a near-Wieferich prime.
%C A258367 p is in A001220 iff a(n) = 0. This is the case iff A014664(n) = A243905(n), which happens for n = 183 and n = 490.
%C A258367 Is a(n) = 0 for any other n, and, if yes, are there infinitely many such n?
%H A258367 Felix Fröhlich, <a href="/A258367/b258367.txt">Table of n, a(n) for n = 2..10000</a>
%H A258367 R. Crandall, K. Dilcher and C. Pomerance, <a href="https://doi.org/10.1090/S0025-5718-97-00791-6">A search for Wieferich and Wilson primes</a>, Mathematics of Computation, 66 (1997), 433-449.
%F A258367 a(n) = min(b(n) mod p, -b(n) mod p) where p = prime(n) and b(n) = Sum_{i=1..ceiling((p-1)/4)} (2i-1)^(p-2). - _Daniel Chen_, Sep 01 2022
%o A258367 (PARI) a(n,p=prime(n))=abs(centerlift(Mod(2,p^2)^((p-1)/2))\/p)
%o A258367 apply(p->a(0,p), primes(100)[2..100]) \\ _Charles R Greathouse IV_, Jun 15 2015
%Y A258367 Cf. A001220, A195988, A241014, A244801, A246568, A250406, A250407, A306885, A338646, A343410.
%K A258367 nonn
%O A258367 2,4
%A A258367 _Felix Fröhlich_, May 28 2015