A318491 a(n) is the numerator of Sum_{d|n} Sum_{j|d} 1/j.

1, 5, 7, 17, 11, 35, 15, 49, 34, 11, 23, 119, 27, 75, 77, 129, 35, 85, 39, 187, 5, 115, 47, 343, 86, 135, 142, 255, 59, 77, 63, 321, 161, 175, 33, 289, 75, 195, 63, 539, 83, 25, 87, 391, 374, 235, 95, 301, 162, 43, 245, 459, 107, 355, 23, 105, 91, 295, 119, 1309, 123, 315, 170, 769, 297

Offset: 1

Author

Ilya Gutkovskiy, Aug 27 2018

Keywords

• nonn
• frac
• look

Examples

			1, 5/2, 7/3, 17/4, 11/5, 35/6, 15/7, 49/8, 34/9, 11/2, 23/11, 119/12, 27/13, 75/14, 77/15, 129/16, ...
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000005, A000203, A006171, A007429, A017665, A017666, A060640, A318492 (denominators).

Cf. A001620, A013661, A306016.

Programs

Maple

f:= proc(n) local d;
   numer(add(numtheory:-sigma(d)/d, d = numtheory:-divisors(n))) end proc:
map(f, [$1..65]); # Robert Israel, Jan 13 2025

Mathematica

Numerator[Table[Sum[DivisorSigma[-1, d], {d, Divisors[n]}], {n, 65}]]
Numerator[Table[Sum[DivisorSigma[1, d]/d, {d, Divisors[n]}], {n, 65}]]
Numerator[Table[Sum[d DivisorSigma[0, d], {d, Divisors[n]}]/n, {n, 65}]]
nmax = 65; Rest[Numerator[CoefficientList[Series[Sum[DivisorSigma[1, k] x^k/(k (1 - x^k)), {k, 1, nmax}], {x, 0, nmax}], x]]]
nmax = 65; Rest[Numerator[CoefficientList[Series[-Log[Product[(1 - x^k)^DivisorSigma[0, k], {k, 1, nmax}]], {x, 0, nmax}], x]]]

PARI

a(n) = numerator(sumdiv(n, d, sumdiv(d, j, 1/j))); \\ Michel Marcus, Aug 28 2018

Formula

Numerators of coefficients in expansion of Sum_{k>=1} sigma(k)*x^k/(k*(1 - x^k)), where sigma(k) = sum of divisors of k (A000203).

Numerators of coefficients in expansion of -log(Product_{k>=1} (1 - x^k)^tau(k)), where tau(k) = number of divisors of k (A000005).

a(n) = numerator of Sum_{d|n} sigma(d)/d.

a(n) = numerator of (1/n)*Sum_{d|n} d*tau(d).

If p is a prime, a(p) = 2*p + 1.

Sum_{k=1..n} a(k)/A318492(k) ~ zeta(2) * n * (log(n) + 2*gamma - 1 + zeta'(2)/zeta(2)), where gamma is Euler's constant (A001620). - Amiram Eldar, Dec 25 2024