A384818 Denominator of the sum of the reciprocals of all square divisors of all positive integers <= n.

1, 1, 1, 4, 4, 4, 4, 2, 18, 18, 18, 36, 36, 36, 36, 144, 144, 144, 144, 144, 144, 144, 144, 144, 3600, 3600, 1200, 1200, 1200, 1200, 1200, 600, 600, 600, 600, 1800, 1800, 1800, 1800, 1800, 1800, 1800, 1800, 1800, 600, 600, 600, 1200, 58800, 58800, 58800, 58800, 58800

Offset: 1

Author

Ilya Gutkovskiy, Jun 10 2025

Keywords

• nonn
• frac

Examples

			1, 2, 3, 17/4, 21/4, 25/4, 29/4, 17/2, 173/18, 191/18, 209/18, 463/36, ...
		

Crossrefs

Cf. A007407, A017668, A284650, A309125, A373440, A384817 (numerators).

Programs

Mathematica

nmax = 53; CoefficientList[Series[1/(1 - x) Sum[x^(k^2)/(k^2 (1 - x^(k^2))), {k, 1, nmax}], {x, 0, nmax}], x] // Denominator // Rest
Table[Sum[Floor[n/k^2]/k^2, {k, 1, Floor[Sqrt[n]]}], {n, 1, 53}] // Denominator

PARI

a(n) = denominator(sum(k=1, n, sumdiv(k, d, if (issquare(d), 1/d)))); \\ Michel Marcus, Jun 10 2025

Formula

G.f. for fractions: (1/(1 - x)) * Sum_{k>=1} x^(k^2) / (k^2*(1 - x^(k^2))).

a(n) is the denominator of Sum_{k=1..floor(sqrt(n))} floor(n/k^2) / k^2.

A384817(n) / a(n) ~ Pi^4 * n / 90.