A178812 -id:A178812 - cp's OEIS frontend

A178813 a(n) = (prime(n)^(p-1) - 1)/p^2 mod p, where p is the first prime that divides (prime(n)^(p-1) - 1)/p.

487, 4, 1, 1, 46, 1, 0, 1, 11, 1, 2, 1, 0, 2

Offset: 1

Jonathan Sondow, Jun 17 2010

a(n) = (prime(n)^(p-1) - 1)/p^2 mod p, where p = A174422(n) is the first Wieferich prime base prime(n).
a(n) = (prime(n)^(p-1) - 1)/p^2 mod p, where p is the first prime such that p^2 divides prime(n)^(p-1) - 1.
See references and additional comments, links, and cross-refs in A001220 and A039951.
a(15) > 2451011, a(16) = 1, a(17) = 4, a(18) = 1, a(19) = 5, a(20) = 2, a(21) = 0, a(22) = 6, a(23) = 1186, a(24) = 0, a(25) = 0, a(26) = 1, a(27) > 10^5, a(28) = 0, a(29) = 1, a(30) = 0, a(31) = 1, a(32) = 7, a(33) = 0, a(35) = 1, a(36) = 4, a(37) = 1, a(38) = 0, a(40) = 1, a(41) = 2, a(42) = 1, a(43) = 2, a(44) = 0, a(45) = 1, a(46) = 2, a(48) = 30, a(49) = 3, a(50) = 1. - J.W.L. (Jan) Eerland, Sep 27 2024

			Prime(2) = 3 and the first prime p that divides (3^(p-1) - 1)/p is 11, so a(2) = (3^10 - 1)/11^2 mod 11 = 488 mod 11 = 4.

Wikipedia, Generalized Wieferich primes

Cf. A001220, A039951, A174422, A178812, A178814.

Table[If[IntegerQ[s[[2]]],s,{s[[1]], "no solution in range 1 <= k <= 10^5"}], {s,Table[k = 1;Monitor[Parallelize[While[k <= 10^5,If[IntegerQ[((Prime[n]^(Prime[k] - 1) - 1)/Prime[k])/Prime[k]],Break[]]; k++];{n, Mod[(Prime[n]^(Prime[k] - 1) - 1)/Prime[k]^2, Prime[k]]}],k], {n, 1, 10}]}] (* J.W.L. (Jan) Eerland, Sep 27 2024 *)

a(n) = k mod 2, if prime(n) = 4k+1.
a(n) = A178814(prime(n)) .
a(1) = A178812(1).

A178814 (n^(p-1) - 1)/p^2 mod p, where p is the first prime that divides (n^(p-1) - 1)/p.

Original entry on oeis.org

0, 487, 4, 974, 1, 30384, 1, 1, 0, 2, 46, 1571, 1, 17, 24160, 855, 0, 4, 1, 189, 1, 5, 11, 1, 0, 0, 1, 0, 1, 3, 2, 3, 0, 19632919407, 1, 60768, 1, 11, 1435, 8, 0, 0, 2, 2, 1, 1

Offset: 1

Jonathan Sondow, Jun 17 2010

(n^(p-1) - 1)/p^2 mod p, where p is the first prime such that p^2 divides n^(p-1) - 1.

See references and additional comments, links, and cross-refs in A001220 and A039951.

			The first prime p that divides (3^(p-1) - 1)/p is 11, so a(3) = (3^10 - 1)/11^2 mod 11 = 488 mod 11 = 4.

Wikipedia, Generalized Wieferich primes

a(2) = A178812(1) = A178813(1). Cf. A001220, A039951, A174422.

a(n) = (n^(p-1) - 1)/p^2 mod p, where p = A039951(n).
a(n) = k mod 2, if n = 4k+1.
a(prime(n)) = A178813(n).

cp's OEIS Frontend

A178813 a(n) = (prime(n)^(p-1) - 1)/p^2 mod p, where p is the first prime that divides (prime(n)^(p-1) - 1)/p.

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