This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A245089 #19 Dec 23 2014 11:26:06
%S A245089 -2,-1,4,-6,8,-6,-10,-5,3,-16,4,6,3,6,-11,-29,2,7,21,4,-16,-23,-5,43,
%T A245089 14,3,-32,26,13,-23,64,52,-30,-74,-17,-33,37,-82,-68,55,-78,96,79,22,
%U A245089 -81,-26,-7,70,-38,9,3,-118,128,-123,-67,-69,-78,-138,30,-60,-19,88,-26,110,27,63,-82,138
%N A245089 The unique integer r with |r| < prime(n)/2 such that B_{prime(n)-2}(1/3) == r (mod prime(n)), where B_m(x) denotes the Bernoulli polynomial of degree m.
%C A245089 Conjecture: a(n) = 0 infinitely often. In other words, there are infinitely many primes p > 3 such that B_{p-2}(1/3) == 0 (mod p).
%C A245089 This seems reasonable in view of the standard heuristic arguments. Our computation shows that if a(n) = 0 then n > 2600 and hence prime(n) > 23000.
%C A245089 Zhi-Wei Sun made many conjectures on congruences involving B_{p-2}(1/3), see the Sci. China Math. paper and arXiv:1407.0967.
%C A245089 The first value of n with a(n) = 0 is 18392. For the prime p = prime(18392) = 205129, we have B_{p-2}(1/3) == 20060*p (mod p^2). - _Zhi-Wei Sun_, Dec 13 2014
%H A245089 Zhi-Wei Sun, <a href="/A245089/b245089.txt">Table of n, a(n) for n = 3..1300</a>
%H A245089 Guo-Shuai Mao and Zhi-Wei Sun, <a href="http://arxiv.org/abs/1412.0523">Two congruences involving harmonic numbers with applications</a>, arXiv:1412.0523 [math.NT], 2014.
%H A245089 Zhi-Wei Sun, <a href="http://dx.doi.org/10.1007/s11425-011-4302-x">Super congruences and Euler numbers</a>, Sci. China Math. 54(2011), 2509-2535.
%H A245089 Zhi-Wei Sun, <a href="http://arxiv.org/abs/1407.0967">Congruences involving g_n(x) = sum_{k=0}^n C(n,k)^2*C(2k,k)*x^k</a>, arXiv:1407.0967 [math.NT], 2014.
%e A245089 a(3) = -2 since B_{prime(3)-2}(1/3) = B_3(1/3) = 1/27 == -2 (mod prime(3)=5).
%t A245089 rMod[m_,n_]:=Mod[Numerator[m]*PowerMod[Denominator[m],-1,n],n,-n/2]
%t A245089 a[n_]:=rMod[BernoulliB[Prime[n]-2,1/3],Prime[n]]
%t A245089 Table[a[n],{n,3,70}]
%Y A245089 Cf. A027641, A027642.
%K A245089 sign
%O A245089 3,1
%A A245089 _Zhi-Wei Sun_, Jul 11 2014