A343410 a(n) is the smallest A (in absolute value) such that for p = prime(n), 3^{(p-1)/2} == +-1 + A*p (mod p^2), i.e., such that p is a base-3 near-Wieferich prime (near-Mirimanoff prime).

1, 2, 3, 0, 4, 5, 9, 3, 7, 7, 9, 10, 1, 18, 15, 9, 24, 26, 16, 13, 27, 32, 12, 45, 8, 10, 49, 4, 2, 30, 28, 9, 47, 6, 22, 47, 49, 50, 56, 43, 66, 55, 22, 14, 74, 9, 61, 96, 21, 25, 47, 22, 111, 64, 23, 5, 114, 128, 110, 121, 86, 56, 90, 156, 117, 48, 166, 133

Offset: 2

Author

Felix Fröhlich, Apr 14 2021

Keywords

• nonn

Comments

a(n) = 0 if and only if p is a base-3 Wieferich prime (Mirimanoff prime, cf. A014127).

These values can be used in a search for Mirimanoff primes to define "near-Mirimanoff primes" by choosing some value x and reporting all primes with |A| <= x in order to get a larger dataset.

Crossrefs

Cf. A014127, A258367.

Programs

PARI

a(n) = my(p=prime(n)); abs(centerlift(Mod(3, p^2)^((p-1)/2))\/p)