A284648 Numerator of sum of reciprocals of all divisors of all positive integers <= n.

1, 5, 23, 67, 407, 527, 4169, 9913, 33379, 7583, 89461, 102397, 1408777, 1532329, 8238221, 17872837, 316811189, 343357709, 6768841271, 7257705647, 7612437167, 7993370447, 189434541721, 202820113921, 1047296788661, 1090542483461, 3390610314383, 3551237180783, 105395281238707

Offset: 1

Author

Ilya Gutkovskiy, Mar 31 2017

Keywords

• nonn
• frac

Comments

The value of (1/n)*Sum_{k=1..n} sigma(k)/k approaches Pi^2/6.

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, ...
		

References

• József Sándor, Dragoslav S. Mitrinović, and Borislav Crstici, Handbook of Number Theory I, Springer, 2006, Section III.5, p. 82.
• Arnold Walfisz, Weylsche Exponentialsummen in der neueren Zahlentheorie, Deutscher Verlag der Wissenschaften, Berlin, 1963, p. 99.

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, A284650 (denominators).

Programs

Maple

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

Mathematica

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

PARI

for(n=1, 29, print1(numerator(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)).numerator 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 a(n)/A284650(n), see example).

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

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

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