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%I #26 Jan 22 2024 11:54:20
%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 The value of (1/n)*Sum_{k=1..n} sigma(k)/k approaches Pi^2/6.
%D József Sándor, Dragoslav S. Mitrinovic, and Borislav Crstici, Handbook of Number Theory I, Springer, 2006, Section III.5, p. 82.
%D Arnold Walfisz, Weylsche Exponentialsummen in der neueren Zahlentheorie, Deutscher Verlag der Wissenschaften, Berlin, 1963, p. 99.
%H Amiram Eldar, <a href="/A284648/b284648.txt">Table of n, a(n) for n = 1..1000</a>
%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.
%F a(n) = numerator of Sum_{k=1..n} floor(n/k)/k. - _Ridouane Oudra_, Jan 21 2024
%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, ...
%p with(numtheory): seq(numer(add(sigma(k)/k, k=1..n)), n=1..40); # _Ridouane Oudra_, Jan 21 2024
%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 sympy import divisor_sigma, Integer
%o 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
%Y Cf. A000203, A017665, A017666, A108775, A284650 (denominators).
%K nonn,frac
%O 1,2
%A _Ilya Gutkovskiy_, Mar 31 2017
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