A178844 - OEIS

%I #2 Mar 30 2012 19:00:09

%S 1,1,3,2,5,3,13,3,17,1,6,1,23,25,44,36,8,36,10,2,56,19,48,6,57,92,59,

%T 13,67,83,18,17,53,30,96,56,82,67,47,3,50,148,50,104,175,135,109,189,

%U 201,68,7,26,142,247,225,128,260,109,70,74,58,78,294,175,120,175,139,153

%N First nonzero Fermat quotient mod the n-th prime.

%C First nonzero value of q_p(m) mod p with gcd(m,p) = 1, where q_p(m) = (m^(p-1) - 1)/p is the Fermat quotient of p to the base m and p is the n-th prime p_n.

%C It is believed that a(n) = q_p(3) mod p, if p = p_n is a Wieferich prime A001220. See Section 1.1 in Ostafe-Shparlinski (2010).

%C See additional comments, references, links, and cross-refs in A001220 and A178815.

%H A. Ostafe and I. Shparlinski (2010), <a href="http://arxiv.org/abs/1001.1504"> Pseudorandomness and Dynamics of Fermat Quotients</a>

%F a(n) = q_p(A178815(n)) mod p, where p = p_n.

%F a(n) = A130912(n), if n > 1 and p_n is not a Wieferich prime. (Note: the offset of A130912 is n = 2.)

%e p_1 = 2 and (m^1 - 1)/2 = 0, 1 == 0, 1 (mod 2) for m = 1, 3, so a(1) = 1.

%e p_5 = 11 and (m^10 - 1)/11 = 0, 93 == 0, 5 (mod 7) for m = 1, 2, so a(4) = 5.

%e p_183 = 1093 and (m^1092 - 1)/1093 == 0, 0, 312 (mod 1093) for m = 1, 2, 3, so a(183) = 312.

%e Similarly, a(490) = 7.

%Y Cf. A001220, A130912, A178815.

%K nonn

%O 1,3

%A _Jonathan Sondow_, Jun 24 2010

%E Nonexistent A-numbers removed by _Jonathan Sondow_, Jun 26 2010