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 A127854 #19 Nov 20 2017 03:25:36
%S A127854 19,61,127,217,331,469,631,817,1027,1261,1519,1801,2107,2437,2791,
%T A127854 3169,3571,3997,4447,4921,5419,5941,6487,7057,7651,8269,8911,9577,
%U A127854 10267,10981,11719,12481,13267,14077,14911,15769,16651,17557,18487,19441
%N A127854 Largest number k such that k^2 divides A007781(6n+1).
%C A127854 A007781(n) = (n+1)^(n+1) - n^n. A007781(6n+1) is not squarefree for n > 0. a(n) is the largest square divisor of A007781(6n+1). All terms belong to A003215 Hex (or centered hexagonal) numbers: 3n(n+1)+1 (crystal ball sequence for hexagonal lattice). It appears that a(n) = A003215(2n) = 6n(2n+1)+1. A007781(6n+1)/A003215(2n)^2 = ((6n+2)^(6n+2)-(6n+1)^(6n+1))/(6n(2n+1)+1)^2 = {44193, 2904899682603, 6378521938392937343349, 128847538453506016002947264859159, 13183819636551142123977274666051092130410345, ...}. Prime terms of a(n) belong to A002407. Factorizations of the terms of a(n) are {19, 61, 127, 7*31, 331, 7*67, 631, 19*43, 13*79, 13*97, 7*7*31, 1801, 7*7*43, 2437, 2791, 3169, 3571, 7*571, 4447, 7*19*37, 5419, 13*457, 13*499, 7067, 7*1093, 8269, 7*19*67, 61*157, 10267, 79*139, ...}. All prime factors of a(n) are of the form 6k+1.
%F A127854 Conjecture: a(n) = 12n^2 + 6n + 1.
%F A127854 Conjecture: a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); g.f.: x*(19 + 4*x + x^2)/(1-x)^3. - _Colin Barker_, Mar 16 2012
%F A127854 These conjectures are false. For n=74, 12*n^2 + 6*n + 1 = 66157 but A007781(6*74+1) is divisible by 5491031^2. - _Robert Israel_, Nov 19 2017
%Y A127854 Cf. A007781 = (n+1)^(n+1) - n^n. Cf. A000312, A068955, A003215, A002407.
%K A127854 nonn
%O A127854 1,1
%A A127854 _Alexander Adamchuk_, Apr 05 2007
%E A127854 a(24) corrected by _T. D. Noe_, Mar 14 2008