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A046094 Agoh's congruence; a(n) is conjectured to be 1 iff n is prime.

%I A046094 #35 Feb 16 2025 08:32:38
%S A046094 0,1,1,0,1,0,1,0,3,0,1,0,1,0,5,0,1,0,1,0,7,0,1,0,5,0,9,0,1,0,1,0,11,0,
%T A046094 0,0,1,0,13,0,1,0,1,0,24,0,1,0,7,0,17,0,1,0,0,0,19,0,1,0,1,0,21,0,13,
%U A046094 0,1,0,23,0,1,0,1,0,25,0,0,0,1,0,27,0,1,0,17,0,29,0,1,0,13,0,31,0,0,0,1,0
%N A046094 Agoh's congruence; a(n) is conjectured to be 1 iff n is prime.
%H A046094 Seiichi Manyama, <a href="/A046094/b046094.txt">Table of n, a(n) for n = 1..1000</a>
%H A046094 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 A046094 Romeo Meštrović, <a href="http://arxiv.org/abs/1305.1867">Generalizations of Carmichael numbers I,</a> arXiv:1305.1867v1 [math.NT], May 4, 2013.
%H A046094 R. Mestrovic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Mestrovic/mes4.html">On a Congruence Modulo n^3 Involving Two Consecutive Sums of Powers</a>, Journal of Integer Sequences, Vol. 17 (2014), 14.8.4.
%H A046094 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/AgohsConjecture.html">Agoh's Conjecture</a>
%H A046094 <a href="/index/Be#Bernoulli">Index entries for sequences related to Bernoulli numbers.</a>
%F A046094 a(n) = - n*Bernoulli(n-1) mod n.
%e A046094 - 21 * Bernoulli(20) = 21 * 174611 / 330 = 1222277 / 110  and 1 / 110 == 17 mod 21, so a(21) = 1222277 * 17 mod 21 = 7. - _Jonathan Sondow_, Aug 13 2013
%t A046094 a[ n_ ] := Mod[ Numerator[ -n* BernoulliB[ n-1 ]]*PowerMod[ Denominator[ n*BernoulliB[ n-1 ] ], -1, n ], n ] (* _Jonathan Sondow_, Aug 13 2013 *)
%o A046094 (PARI) a(n) = -n*bernfrac(n-1) % n; \\ _Michel Marcus_, Aug 08 2017
%Y A046094 Cf. A228037.
%K A046094 nonn
%O A046094 1,9
%A A046094 _Eric W. Weisstein_
%E A046094 a(21), a(51), a(57), a(65), a(81) corrected by _Jonathan Sondow_, Aug 13 2013