A179400 Primes which are the fourth element of a generalized Wieferich sequence.

331, 359, 1549, 1777, 2011, 6211, 7481, 10369, 13477, 19069, 20431, 22567, 28289, 32933, 39041, 40597, 77713, 96979, 101489

Offset: 1

Author

M. F. Hasler, Jan 10 2011

Keywords

• nonn
• hard
• more

Comments

A generalized Wieferich sequence is an increasing sequence of primes p[1],p[2],... such that a=p[n+1] is a Wieferich prime to base b=p[n], i.e., a^(b-1)=1 (mod b^2).

Examples

			The smallest generalized Wieferich sequence of length 4 is (3,11,71,331): 3^10=1 (mod 11^2), 11^70=1 (mod 71^2), 71^330=1 (mod 331^2). Therefore, a(1)=331.
This can actually be extended with 359 to such a sequence of length 5, since 331^358=1 (mod 359^2). Therefore, the next such sequence is (11,71,331,359) and a(2)=359.
Then comes a(3)=1549 from the sequence (307, 487, 1069, 1549). Note that this sequence "starts later" than (197, 653, 1381, 1777) which yields a(4)=1777.
		

Links

• Kevin Acres, Mike Oakes, David Broadhurst, Makoto Kamada, 1993/2011 puzzle, digest of 15 messages in primenumbers Yahoo group, Jan 8 - Jan 9, 2011.
• D. Broadhurst, Re: 1993/2011 puzzle [and Puzzle 7], in [e-mail], Jan 2011.

Crossrefs

Cf. A001220, A174422 and references therein.

Programs

PARI

fp(p)={ my(a=precprime(p)); while(Mod(a,p^2)^(p-1)-1 && a=precprime(a-1),);a }
forprime(p=1,default(primelimit),my(a=p); for(c=1,3,(a=fp(a))||next(2));print1(p, ", "))