%I #4 Mar 30 2012 17:22:50

%S 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,

%T 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,

%U 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

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

%C 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.

%H T. D. Noe, <a href="/A143811/b143811.txt">Table of n, a(n) for n=1..10000</a>

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

%K nonn

%O 1,5

%A _T. D. Noe_, Sep 02 2008