A357556 a(n) is the denominator of Sum_{d|n} (-1)^(d+1) / d^2.

1, 4, 9, 16, 25, 6, 49, 64, 81, 50, 121, 72, 169, 98, 45, 256, 289, 108, 361, 200, 441, 242, 529, 288, 625, 338, 729, 392, 841, 15, 961, 1024, 1089, 578, 49, 1296, 1369, 722, 1521, 800, 1681, 147, 1849, 88, 2025, 1058, 2209, 128, 2401, 2500, 2601, 1352, 2809, 243, 3025

Offset

1,2

Links

Table of n, a(n) for n=1..55.

Formula

Denominators of coefficients in expansion of Sum_{k>=1} (-1)^(k+1) * x^k / (k^2 * (1 - x^k)).

Example

1, 3/4, 10/9, 11/16, 26/25, 5/6, 50/49, 43/64, 91/81, 39/50, 122/121, ...

Mathematica

Table[Sum[(-1)^(d + 1)/d^2, {d, Divisors[n]}], {n, 1, 55}] // Denominator
nmax = 55; CoefficientList[Series[Sum[(-1)^(k + 1) x^k/(k^2 (1 - x^k)), {k, 1, nmax}], {x, 0, nmax}], x] // Denominator // Rest

Prog

(Python)
from sympy import divisors
from fractions import Fraction
def a(n): return sum(Fraction((-1)**(d+1), d*d) for d in divisors(n, generator=True)).denominator
print([a(n) for n in range(1, 56)]) # Michael S. Branicky, Oct 03 2022
(PARI) a(n) = denominator(sumdiv(n, d, (-1)^(d+1)/d^2)); \\ Michel Marcus, Oct 03 2022

Crossrefs

Cf. A017668, A064027, A098988, A321543, A334580, A357555 (numerators).

Sequence in context: A070452 A070653 A304233 * A070451 A070450 A070449

Adjacent sequences: A357553 A357554 A357555 * A357557 A357558 A357559

Keyword

nonn,frac

Author

Ilya Gutkovskiy, Oct 03 2022

Status

approved