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A235140 Numerator(m*Bernoulli(m-1)+1) (mod m), for m = 1, 3, 5, 7, 9, ...

%I A235140 #18 Feb 16 2025 08:33:21
%S A235140 0,0,0,0,7,0,0,7,0,0,12,0,16,11,0,0,16,6,0,15,0,0,22,0,8,5,0,28,24,0,
%T A235140 0,23,11,0,56,0,0,27,30,0,71,0,63,31,0,69,36,6,0,35,0,0,50,0,0,99,0,
%U A235140 42,44,6,72,43,106,0,84,0,1,47,0,0,91,6,36,51,0,0,112,138,0,55,102,0,78,0,115,136,0,79,67,0,0,63,23,42,136,0,0,67,0,0,111
%N A235140 Numerator(m*Bernoulli(m-1)+1) (mod m), for m = 1, 3, 5, 7, 9, ...
%H A235140 Robert Israel, <a href="/A235140/b235140.txt">Table of n, a(n) for n = 0..6070</a>
%H A235140 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GiugasConjecture.html">Giuga's Conjecture</a>
%H A235140 Wikipedia, <a href="http://en.wikipedia.org/wiki/Agoh-Giuga_conjecture">Agoh-Giuga conjecture</a>
%F A235140 a(n) = numerator((2*n+1)*Bernoulli(2*n)+1) (mod 2*n+1).
%F A235140 The Agoh-Giuga Conjecture is that a(n)=0 iff 2*n+1 is 1 or a prime.
%F A235140 a(n) = 0 iff A235363(n) = 0.
%F A235140 Conjecture: a(n) = (2*n + 7)/3 if n > 1 is in A004611. - _Robert Israel_, Nov 07 2024
%p A235140 seq(numer(m*bernoulli(m-1)+1) mod m, m = 1 .. 300, 2); # _Robert Israel_, Nov 07 2024
%t A235140 Table[ Mod[ Numerator[ n*BernoulliB[n - 1] + 1], n], {n, 1, 201, 2}]
%Y A235140 Cf. A004611, A007850, A046094, A204187, A228037, A235363.
%K A235140 nonn,look
%O A235140 0,5
%A A235140 _Jonathan Sondow_, Jan 07 2014