A174422 1st Wieferich prime base prime(n).

1093, 11, 2, 5, 71, 2, 2, 3, 13, 2, 7, 2, 2, 5

Offset: 1

Author

Jonathan Sondow, Mar 19 2010

Keywords

• hard
• more
• nonn

Comments

Smallest prime p such that p^2 divides prime(n)^(p-1) - 1.

Smallest prime p such that p divides the Fermat quotient q_p((prime(n)) = (prime(n)^(p-1) - 1)/p.

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

a(15) = A039951(47) > 4.1*10^13.

Examples

			a(1) = 1093 is the first Wieferich prime A001220. a(2) = 11 is the first Mirimanoff prime A014127.
		

Links

• Wikipedia, Generalized Wieferich primes.
• Robert G. Wilson v, Table of n, a(n) for n = 1..10000 (or 0 if unknown)

Crossrefs

Cf. A001220, A014127, A039951 = smallest prime p such that p^2 divides n^(p-1) - 1, A125636 = smallest prime p such that prime(n)^2 divides p^(prime(n)-1) - 1.

Cf. A178871 = 2nd Wieferich prime base prime(n).

Programs

Mathematica

f[n_] := Block[{b = Prime@ n, p = 2}, While[ PowerMod[b, p - 1, p^2] != 1, p = NextPrime@ p]; p]; Array[f, 14]

PARI

forprime(a=2, 20, forprime(p=2, 10^9, if(Mod(a, p^2)^(p-1)==1, print1(p, ", "); next({2}))); print1("--, ")) \\ Felix Fröhlich, Jun 27 2014

Formula

a(n) = A039951(prime(n)).

a(n) = 2 if and only if prime(n) == 1 (mod 4). [Jonathan Sondow, Aug 29 2010]