

A179400


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


3



331, 359, 1549, 1777, 2011, 6211, 7481, 10369, 13477, 19069, 20431, 22567, 28289, 32933, 39041, 40597, 77713, 96979, 101489
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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^(b1)=1 (mod b^2).


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.


EXAMPLE

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.


PROG

(PARI) fp(p)={ my(a=precprime(p)); while(Mod(a, p^2)^(p1)1 & a=precprime(a1), ); a }
forprime(p=1, default(primelimit), a=p; for(c=1, 3, (a=fp(a))next(2)); print1(p", "))


CROSSREFS



KEYWORD

nonn,hard,more


AUTHOR



STATUS

approved



