A384817 Numerator of the sum of the reciprocals of all square divisors of all positive integers <= n.

1, 2, 3, 17, 21, 25, 29, 17, 173, 191, 209, 463, 499, 535, 571, 2473, 2617, 2777, 2921, 3101, 3245, 3389, 3533, 3713, 96569, 100169, 34723, 36223, 37423, 38623, 39823, 20699, 21299, 21899, 22499, 69997, 71797, 73597, 75397, 77647, 79447, 81247, 83047, 85297

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. A007406, A017667, A284648, A309125, A373439, A384818 (denominators).

Programs

Mathematica

nmax = 44; CoefficientList[Series[1/(1 - x) Sum[x^(k^2)/(k^2 (1 - x^(k^2))), {k, 1, nmax}], {x, 0, nmax}], x] // Numerator // Rest
Table[Sum[Floor[n/k^2]/k^2, {k, 1, Floor[Sqrt[n]]}], {n, 1, 44}] // Numerator

PARI

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

a(n) / A384818(n) ~ Pi^4 * n / 90.