Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #13 Feb 15 2016 17:07:41
%S 0,6,5,42,22,13,102,57,207,551,620,296,697,602,329,1855,1652,3477,871,
%T 4970,876,5846,1743,6319,6887,7373,5974,214,3379,10848,9144,15720,
%U 7809,9452,14155,13137,23864,17767,3674,18511,8771,13213,30560,6948,29156,23681
%N p*B_(p-1)+1 modulo p^2, where p = prime(n) and B_i denotes the i-th Bernoulli number.
%C Related to the Agoh-Giuga conjecture (called Agoh's conjecture by Borwein, Borwein, Borwein and Girgensohn) which states that a positive integer k is prime if and only if k*B_(k-1) == -1 (mod k) (see Wikipedia and Borwein, Borwein, Borwein, Girgensohn, 1996, open problem 10).
%C Up to p = 101839, there are only two primes p such that a(n) = 0, namely 2 and 1277, i.e., a(1) = 0 and a(206) = 0. Do any other such primes exist?
%H Felix Fröhlich, <a href="/A268000/b268000.txt">Table of n, a(n) for n = 1..1000</a>
%H D. Borwein, J. M. Borwein, P. B. Borwein and R. Girgensohn, <a href="http://dx.doi.org/10.2307/2975213">Giuga's conjecture on primality</a>, The American Mathematical Monthly, Vol. 103, No. 1 (1996), 40-50.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Agoh-Giuga_conjecture">Agoh-Giuga conjecture</a>
%o (PARI) a(n) = my(p=prime(n)); lift(Mod(p*bernfrac(p-1)+1, p^2))
%K nonn
%O 1,2
%A _Felix Fröhlich_, Jan 24 2016