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A284648
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Numerator of sum of reciprocals of all divisors of all positive integers <= n.
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4
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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
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OFFSET
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1,2
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COMMENTS
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The value of (1/n)*Sum_{k=1..n} sigma(k)/k approaches Pi^2/6.
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REFERENCES
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József Sándor, Dragoslav S. Mitrinovic, 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.
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LINKS
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FORMULA
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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
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EXAMPLE
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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, ...
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MAPLE
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with(numtheory): seq(numer(add(sigma(k)/k, k=1..n)), n=1..40); # Ridouane Oudra, Jan 21 2024
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MATHEMATICA
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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]]]
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,frac
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AUTHOR
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STATUS
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approved
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