A284650 Denominator of sum of reciprocals of all divisors of all positive integers <= n.

1, 2, 6, 12, 60, 60, 420, 840, 2520, 504, 5544, 5544, 72072, 72072, 360360, 720720, 12252240, 12252240, 232792560, 232792560, 232792560, 232792560, 5354228880, 5354228880, 26771144400, 26771144400, 80313433200, 80313433200, 2329089562800

Offset: 1

Author

Ilya Gutkovskiy, Mar 31 2017

Keywords

• nonn
• frac

Examples

			1, 5/2, 23/6, 67/12, 407/60, 527/60, 4169/420, 9913/840, 33379/2520, 7583/504, 89461/5544, 102397/5544, 1408777/72072, 1532329/72072, 8238221/360360, ...
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..1000
• Index entries for sequences related to sums of divisors

Crossrefs

Cf. A000203, A017665, A017666, A108775, A284648 (numerators).

Programs

Maple

with(numtheory): seq(denom(add(sigma(k)/k, k=1..n)), n=1..40); # Ridouane Oudra, Jan 21 2024

Mathematica

Table[Denominator[Sum[DivisorSigma[-1, k], {k, 1, n}]], {n, 1, 29}]
Table[Denominator[Sum[DivisorSigma[1, k]/k, {k, 1, n}]], {n, 1, 29}]
nmax = 29; Rest[Denominator[CoefficientList[Series[1/(1 - x) Sum[Log[1/(1 - x^k)], {k, 1, nmax}], {x, 0, nmax}], x]]]

PARI

for(n=1, 29, print1(denominator(sum(k=1, n, sigma(k)/k)),", ")) \\ Indranil Ghosh, Mar 31 2017

Python

from sympy import divisor_sigma, Integer
print([sum(divisor_sigma(k)/Integer(k) for k in range(1, n + 1)).denominator for n in range(1, 30)]) # Indranil Ghosh, Mar 31 2017

Formula

G.f.: (1/(1 - x))*Sum_{k>=1} log(1/(1 - x^k)) (for A284648(n)/a(n), see example).

a(n) = denominator of Sum_{k=1..n} Sum_{d|k} 1/d.

a(n) = denominator of Sum_{k=1..n} sigma(k)/k.

a(n) = denominator of Sum_{k=1..n} floor(n/k)/k. - Ridouane Oudra, Jan 21 2024