A143811 Number of numbers k<p such that k^(p-1)-1 is divisible by p^2, p = prime(n).

1, 1, 1, 1, 3, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 1, 3, 1, 2, 1, 1, 2, 1, 2, 1, 2, 2, 3, 3, 2, 1, 1, 2, 1, 3, 1, 1, 1, 2, 2, 1, 2, 2, 3, 2, 1, 3, 2, 1, 2, 1, 2, 2, 4, 1, 1, 2, 2, 2, 2, 1, 3, 1, 4, 1, 3, 3, 3, 3, 3, 2, 2, 1, 1, 2, 2, 1, 2, 2, 1, 2, 2, 1, 2, 1, 2, 3, 1, 1, 5, 1, 2, 1, 3, 2, 2, 1, 2, 2, 2, 1, 4

Offset

1,5

Comments

Note that a(n)>0 because k=1 is always a solution. The primes for which a(n)>1 are given in A134307. The values of k are the terms <p in row n of A143548. The largest known terms in this sequence are for the Wieferich primes 1093 and 3511, for which we have a(183)=11 and a(490)=12, respectively. It is not hard to show that k=p-1 is never a solution for odd prime p. In fact, (p-1)^(p-1)=p+1 (mod p^2) for odd prime p.

Links

T. D. Noe, Table of n, a(n) for n=1..10000

Mathematica

Table[p=Prime[n]; s=Select[Range[p-1], PowerMod[ #, p-1, p^2]==1&]; Length[s], {n, 100}]

Crossrefs

Sequence in context: A328391 A109393 A030348 * A109673 A023591 A165661

Adjacent sequences: A143808 A143809 A143810 * A143812 A143813 A143814

Keyword

nonn

Author

T. D. Noe, Sep 02 2008

Status

approved