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 A284648 Numerator of sum of reciprocals of all divisors of all positive integers <= n. 1

%I

%S 1,5,23,67,407,527,4169,9913,33379,7583,89461,102397,1408777,1532329,

%T 8238221,17872837,316811189,343357709,6768841271,7257705647,

%U 7612437167,7993370447,189434541721,202820113921,1047296788661,1090542483461,3390610314383,3551237180783,105395281238707

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

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

%H <a href="/index/Su#sums_of_divisors">Index entries for sequences related to sums of divisors</a>

%F G.f.: (1/(1 - x))*Sum_{k>=1} log(1/(1 - x^k)) (for a(n)/A284650(n), see example).

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

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

%e 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, ...

%t Table[Numerator[Sum[DivisorSigma[-1, k], {k, 1, n}]], {n, 1, 29}]

%t Table[Numerator[Sum[DivisorSigma[1, k]/k, {k, 1, n}]], {n, 1, 29}]

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

%o (PARI) for(n=1, 29, print1(numerator(sum(k=1, n, sigma(k)/k)),", ")) \\ _Indranil Ghosh_, Mar 31 2017

%o (Python)

%o from fractions import Fraction

%o from sympy import divisor_sigma

%o 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

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

%K nonn,frac

%O 1,2

%A _Ilya Gutkovskiy_, Mar 31 2017

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Last modified January 23 17:24 EST 2019. Contains 319399 sequences. (Running on oeis4.)