A179678 Primes which are the fifth element of a generalized Wieferich sequence.

359, 6211, 13477, 19069, 20431, 22567, 32933, 39041

Offset

1,1

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).

The present sequence is a subsequence of A179400. The numbers 728437, 1051139, 144 449603 and 5176 948723 found by M. Kamada and D. Broadhurst are also in this sequence. The last two numbers are even 7th elements of a generalized Wieferich sequence, cf. the link.

Links

Table of n, a(n) for n=1..8.

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 primenumbers@yahoogroups.com, Jan 2011.

Example

The smallest generalized Wieferich sequence of length 5 is (3, 11, 71, 331, 359): 3^10=1 (mod 11^2), 11^70=1 (mod 71^2), 71^330=1 (mod 331^2) and 331^358=1 (mod 359^2). Therefore, a(1)=359.
It happens that after a(2)=6211, there are four subsequent elements of A179400, namely 13477, 19069, 20431 and 22567, which are also member of a sequence of length 5 and thus in this sequence.

Prog

(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, 4, (a=fp(a))||next(2)); print1(p, ", "))

Crossrefs

Sequence in context: A158307 A013325 A344284 * A101996 A237017 A360943

Adjacent sequences: A179675 A179676 A179677 * A179679 A179680 A179681

Keyword

nonn

Author

M. F. Hasler, Jan 10 2011

Status

approved