A178871 2nd Wieferich prime base prime(n).

3511, 1006003, 20771, 491531

Offset: 1

Author

Jonathan Sondow, Jun 20 2010, Jun 24 2010

Keywords

• hard
• more
• nonn

Comments

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

2nd prime p such that p divides the Fermat quotient q_p(p_n) = ((p_n)^(p-1) - 1)/p, where p_n = prime(n).

a(5) is unknown: 71 is the only known prime p that divides q_p(11).

If a(5) is found, the sequence continues a(6) = 863, a(7) = 3, a(8) = 7, a(9) = 2481757.

See additional comments, references, links, and cross-refs in A039951 and A174422.

Examples

			a(1) = 3511 is the 2nd Wieferich prime A001220(2).
a(2) = 1006003 is the 2nd Mirimanoff prime A014127(2).
		

Links

• Wikipedia, Generalized Wieferich primes

Crossrefs

Cf. A039951 = smallest prime p such that p^2 divides n^(p-1) - 1, A174422 = first Wieferich prime base prime(n).

Programs

PARI

{default(primelimit, 10^7); for(n=1, 9, a=prime(n); c=0; forprime(p=2, 10^7, if(Mod(a, p^2)^(p-1)==1, c++; if(c==2, print1(p, ", "); next(2)))); print1(">10^7, "))} \\ Jens Kruse Andersen, Jun 18 2014