A268000 p*B_(p-1)+1 modulo p^2, where p = prime(n) and B_i denotes the i-th Bernoulli number.

0, 6, 5, 42, 22, 13, 102, 57, 207, 551, 620, 296, 697, 602, 329, 1855, 1652, 3477, 871, 4970, 876, 5846, 1743, 6319, 6887, 7373, 5974, 214, 3379, 10848, 9144, 15720, 7809, 9452, 14155, 13137, 23864, 17767, 3674, 18511, 8771, 13213, 30560, 6948, 29156, 23681

Offset

1,2

Comments

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).

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?

Links

Felix Fröhlich, Table of n, a(n) for n = 1..1000

D. Borwein, J. M. Borwein, P. B. Borwein and R. Girgensohn, Giuga's conjecture on primality, The American Mathematical Monthly, Vol. 103, No. 1 (1996), 40-50.

Wikipedia, Agoh-Giuga conjecture

Prog

(PARI) a(n) = my(p=prime(n)); lift(Mod(p*bernfrac(p-1)+1, p^2))

Crossrefs

Sequence in context: A038259 A358590 A302750 * A223529 A189422 A266980

Adjacent sequences: A267997 A267998 A267999 * A268001 A268002 A268003

Keyword

nonn

Author

Felix Fröhlich, Jan 24 2016

Status

approved