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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). 0
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 (list; graph; refs; listen; history; text; internal format)
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
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
KEYWORD
nonn
AUTHOR
EXTENSIONS
More terms from Michel Marcus, Feb 17 2016
STATUS
approved

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Last modified July 12 07:52 EDT 2024. Contains 374239 sequences. (Running on oeis4.)