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 A202533 #38 Feb 17 2025 03:17:44
%S A202533 2,6,10,42,2,22,130,10,34,798,70,230,2,6,58,4774,154,14,962,26,82,602,
%T A202533 42,658,34,374,5830,6270,38,118,7930,390,26,3082,46,7810,1606,22,22,
%U A202533 158,306,2822,12818,754,2314,168454,12958,418,2134,66,1010,7210,14,214,104858
%N A202533 For a polynomial P(m) with rational coefficients, denote by lcmd(P) the LCM of the denominators of all its coefficients. Then a(n) = lcmd(Sum_{i=1..m} (i^n*Sum_{j=1..i} j^n))/ lcmd((Sum_{i=1..m} i^n)^2).
%C A202533 Note also that Sum_{i=1..m} i^n is a polynomial in m of degree n+1.
%C A202533 One can prove that all terms of the sequence are integers.
%C A202533 The sequence appears to possess an astonishing property: every odd prime p is the maximal prime divisor of a((p-1)/2).
%e A202533 Let n=1. Since lcmd(Sum_{i=1..m} i) = 2, lcmd(Sum_{i=1..m} i^2) = 6, lcmd(Sum_{i=1..m} i^3) = 4, then lcmd(Sum_{i=1..m} i*Sum_{j=1..i} j) = lcmd(Sum_{i=1..m} i^2*(i+1)/2) = 24, therefore, a(1) = 24/4 = 6.
%e A202533 Let p=53. Then a(26) = 5830 = 2*5*11*53 has maximal prime divisor 53.
%t A202533 LCMD[P_, m_] := LCM @@ Denominator[CoefficientList[P // FunctionExpand, m] ]; a[n_] := LCMD[Sum[i^n*HarmonicNumber[i, -n], {i, 1, m}], m]/ LCMD[ HarmonicNumber[m, -n]^2, m]; Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 0, 60}] (* _Jean-François Alcover_, Feb 18 2016 *)
%o A202533 (PARI) sp(p) = x * Polrev(vector(p+1, k, (-1)^(k==p)*binomial(p+1, k)*bernfrac(p+1-k))/(p+1));
%o A202533 lcmd(pol) = lcm(apply(x->denominator(x), Vec(pol)));
%o A202533 a(n) = {pol = x^n*sp(n); pnum = sum(i=0, poldegree(pol), polcoeff(pol, i)*sp(i)); lcmd(pnum)/lcmd(sp(n)^2);} \\ _Michel Marcus_, Feb 17 2016
%Y A202533 Cf. A064538.
%K A202533 nonn
%O A202533 0,1
%A A202533 _Vladimir Shevelev_ and _Peter J. C. Moses_, Dec 20 2011
%E A202533 More terms from _Michel Marcus_, Feb 17 2016