A285931 Number of primes q < p such that q^(p-1) == 1 (modulo p^2), where p = prime(n).

0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0

Offset: 1

Author

Felix Fröhlich, Apr 30 2017

Keywords

• nonn

Comments

Pairs of prime numbers (q, p) satisfying the conditions in the definition are sometimes called "Wieferich prime pairs" (cf. Mossinghoff, 2009).

a(n) > 0 iff p is a term of A222184.

First occurrence of k beginning at 0: 1, 5, 70, 1618, 2702, etc. - Robert G. Wilson v, May 10 2017

Examples

			For n = 70: prime(70) = 349 and there are two primes q < 349 such that q^(349-1) == 1 (modulo 349^2), namely 223 and 317, so a(70) = 2.
		

Links

• M. J. Mossinghoff, Wieferich pairs and Barker sequences, Designs, Codes and Cryptography, Vol. 53, No. 3 (2009), 149-163.

Crossrefs

Cf. A222184, A222206 (records).

Programs

Mathematica

f[n_] := Block[{c = 0, p = Prime@ n, q = 2}, While[q < p, If[ PowerMod[q, p - 1, p^2] == 1, c++]; q = NextPrime@q]; c]; Array[f, 105] (* Robert G. Wilson v, May 10 2017 *)

PARI

a(n) = my(p=prime(n), i=0); forprime(q=1, p-1, if(Mod(q, p^2)^(p-1)==1, i++)); i