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).

2, 6, 10, 42, 2, 22, 130, 10, 34, 798, 70, 230, 2, 6, 58, 4774, 154, 14, 962, 26, 82, 602, 42, 658, 34, 374, 5830, 6270, 38, 118, 7930, 390, 26, 3082, 46, 7810, 1606, 22, 22, 158, 306, 2822, 12818, 754, 2314, 168454, 12958, 418, 2134, 66, 1010, 7210, 14, 214, 104858

Offset

0,1

Comments

Note also that Sum_{i=1..m} i^n is a polynomial in m of degree n+1.

One can prove that all terms of the sequence are integers.

The sequence appears to possess an astonishing property: every odd prime p is the maximal prime divisor of a((p-1)/2).

Links

Table of n, a(n) for n=0..54.

Example

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.
Let p=53. Then a(26) = 5830 = 2*5*11*53 has maximal prime divisor 53.

Mathematica

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 *)

Prog

(PARI) sp(p) = x * Polrev(vector(p+1, k, (-1)^(k==p)*binomial(p+1, k)*bernfrac(p+1-k))/(p+1));
lcmd(pol) = lcm(apply(x->denominator(x), Vec(pol)));
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

Crossrefs

Cf. A064538.

Sequence in context: A027165 A111414 A308486 * A275700 A258899 A248784

Adjacent sequences: A202530 A202531 A202532 * A202534 A202535 A202536

Keyword

nonn

Author

Vladimir Shevelev and Peter J. C. Moses, Dec 20 2011

Extensions

More terms from Michel Marcus, Feb 17 2016

Status

approved