A357555 a(n) is the numerator of Sum_{d|n} (-1)^(d+1) / d^2.

1, 3, 10, 11, 26, 5, 50, 43, 91, 39, 122, 55, 170, 75, 52, 171, 290, 91, 362, 143, 500, 183, 530, 215, 651, 255, 820, 275, 842, 13, 962, 683, 1220, 435, 52, 1001, 1370, 543, 1700, 559, 1682, 125, 1850, 61, 2366, 795, 2210, 95, 2451, 1953, 2900, 935, 2810, 205, 3172

Offset

1,2

Links

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

Formula

Numerators 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}] // Numerator
nmax = 55; CoefficientList[Series[Sum[(-1)^(k + 1) x^k/(k^2 (1 - x^k)), {k, 1, nmax}], {x, 0, nmax}], x] // Numerator // 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)).numerator
print([a(n) for n in range(1, 56)]) # Michael S. Branicky, Oct 03 2022
(PARI) a(n) = numerator(sumdiv(n, d, (-1)^(d+1)/d^2)); \\ Michel Marcus, Oct 03 2022

Crossrefs

Cf. A017667, A064027, A098987, A119682, A321543, A357556 (denominators).

Sequence in context: A352776 A255160 A344892 * A078306 A136815 A119223

Adjacent sequences: A357552 A357553 A357554 * A357556 A357557 A357558

Keyword

nonn,frac

Author

Ilya Gutkovskiy, Oct 03 2022

Status

approved