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A284648 Numerator of sum of reciprocals of all divisors of all positive integers <= n. 1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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

LINKS

Table of n, a(n) for n=1..29.

Index entries for sequences related to sums of divisors

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.

EXAMPLE

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, ...

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]]]

PROG

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

(Python)

from fractions import Fraction

from sympy import divisor_sigma

print [Fraction(str(sum([divisor_sigma(k)/k for k in xrange(1, n + 1)]))).numerator for n in xrange(1, 30)] # Indranil Ghosh, Mar 31 2017

CROSSREFS

Cf. A000203, A017665, A017666, A108775, A284650 (denominators).

Sequence in context: A241765 A106956 A084671 * A243442 A064395 A230497

Adjacent sequences:  A284645 A284646 A284647 * A284649 A284650 A284651

KEYWORD

nonn,frac

AUTHOR

Ilya Gutkovskiy, Mar 31 2017

STATUS

approved

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Last modified February 25 22:49 EST 2018. Contains 299662 sequences. (Running on oeis4.)