|
%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
|
