cp's OEIS Frontend

A284650 Denominator of sum of reciprocals of all divisors of all positive integers <= n.

%I A284650 #24 Jun 27 2025 10:53:14
%S A284650 1,2,6,12,60,60,420,840,2520,504,5544,5544,72072,72072,360360,720720,
%T A284650 12252240,12252240,232792560,232792560,232792560,232792560,5354228880,
%U A284650 5354228880,26771144400,26771144400,80313433200,80313433200,2329089562800
%N A284650 Denominator of sum of reciprocals of all divisors of all positive integers <= n.
%H A284650 Amiram Eldar, <a href="/A284650/b284650.txt">Table of n, a(n) for n = 1..1000</a>
%H A284650 <a href="/index/Su#sums_of_divisors">Index entries for sequences related to sums of divisors</a>
%F A284650 G.f.: (1/(1 - x))*Sum_{k>=1} log(1/(1 - x^k)) (for A284648(n)/a(n), see example).
%F A284650 a(n) = denominator of Sum_{k=1..n} Sum_{d|k} 1/d.
%F A284650 a(n) = denominator of Sum_{k=1..n} sigma(k)/k.
%F A284650 a(n) = denominator of Sum_{k=1..n} floor(n/k)/k. - _Ridouane Oudra_, Jan 21 2024
%e A284650 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 A284650 with(numtheory): seq(denom(add(sigma(k)/k, k=1..n)), n=1..40); # _Ridouane Oudra_, Jan 21 2024
%t A284650 Table[Denominator[Sum[DivisorSigma[-1, k], {k, 1, n}]], {n, 1, 29}]
%t A284650 Table[Denominator[Sum[DivisorSigma[1, k]/k, {k, 1, n}]], {n, 1, 29}]
%t A284650 nmax = 29; Rest[Denominator[CoefficientList[Series[1/(1 - x) Sum[Log[1/(1 - x^k)], {k, 1, nmax}], {x, 0, nmax}], x]]]
%o A284650 (PARI) for(n=1, 29, print1(denominator(sum(k=1, n, sigma(k)/k)),", ")) \\ _Indranil Ghosh_, Mar 31 2017
%o A284650 (Python)
%o A284650 from sympy import divisor_sigma, Integer
%o A284650 print([sum(divisor_sigma(k)/Integer(k) for k in range(1, n + 1)).denominator for n in range(1, 30)]) # _Indranil Ghosh_, Mar 31 2017
%Y A284650 Cf. A000203, A017665, A017666, A108775, A284648 (numerators).
%K A284650 nonn,frac
%O A284650 1,2
%A A284650 _Ilya Gutkovskiy_, Mar 31 2017