A229833 - OEIS

%I #13 Nov 08 2014 09:31:18

%S 17,1175,67232195,1282936297603,171594913930219489,

%T 368517627392700495869,259067037992493907740808871,

%U 63098504840897942292160460526547792021,4948605372033572359620687688871811178548595,169413083241708480729625174442441002390094469490644564301,90165569601996395473034926239938857618854516797194687641929891

%N 1/p^3 * numerator((sum_{j=1..p-1} j^(p-1)) - p*Bernoulli(p-1)) with p = prime(n).

%C Sum_{j=1..p-1} j^(p-1)) == p*Bernoulli(p-1) (mod p^3) for prime p > 3 (see formulas (8) and (10) in "Lerch Quotients, ..."), so a(n) is an integer for n > 2.

%H J. Sondow, <a href="http://arxiv.org/abs/1110.3113">Lerch Quotients, Lerch Primes, Fermat-Wilson Quotients, and the Wieferich-non-Wilson Primes 2, 3, 14771</a>, in Proceedings of CANT 2011, arXiv:1110.3113

%H J. Sondow, <a href="http://link.springer.com/chapter/10.1007%2F978-1-4939-1601-6_17">Lerch Quotients, Lerch Primes, Fermat-Wilson Quotients, and the Wieferich-non-Wilson Primes 2, 3, 14771</a>, Combinatorial and Additive Number Theory, CANT 2011 and 2012, Springer Proc. in Math. & Stat., vol. 101 (2014), pp. 243-255.

%H <a href="/index/Be#Bernoulli">Index entries for sequences related to Bernoulli numbers.</a>

%e Prime(3) = 5 and 1/5^3 * numerator((sum_{j=1..4} j^4) - 5*Bernoulli(4)) = 1/125 * numerator(354 - 5*(-1/6)) = 2125/125 = 17, so a(3) = 17.

%t Table[p = Prime[n]; Numerator[ Sum[j^(p - 1), {j, 1, p - 1}] - p*BernoulliB[p - 1]]/p^3, {n, 3, 13}]

%Y Cf. A197630.

%K nonn

%O 3,1

%A _Jonathan Sondow_, Oct 16 2013