cp's OEIS Frontend

A318491 a(n) is the numerator of Sum_{d|n} Sum_{j|d} 1/j.

%I A318491 #17 Jan 14 2025 01:50:18
%S A318491 1,5,7,17,11,35,15,49,34,11,23,119,27,75,77,129,35,85,39,187,5,115,47,
%T A318491 343,86,135,142,255,59,77,63,321,161,175,33,289,75,195,63,539,83,25,
%U A318491 87,391,374,235,95,301,162,43,245,459,107,355,23,105,91,295,119,1309,123,315,170,769,297
%N A318491 a(n) is the numerator of Sum_{d|n} Sum_{j|d} 1/j.
%H A318491 Robert Israel, <a href="/A318491/b318491.txt">Table of n, a(n) for n = 1..10000</a>
%F A318491 Numerators of coefficients in expansion of Sum_{k>=1} sigma(k)*x^k/(k*(1 - x^k)), where sigma(k) = sum of divisors of k (A000203).
%F A318491 Numerators of coefficients in expansion of -log(Product_{k>=1} (1 - x^k)^tau(k)), where tau(k) = number of divisors of k (A000005).
%F A318491 a(n) = numerator of Sum_{d|n} sigma(d)/d.
%F A318491 a(n) = numerator of (1/n)*Sum_{d|n} d*tau(d).
%F A318491 If p is a prime, a(p) = 2*p + 1.
%F A318491 Sum_{k=1..n} a(k)/A318492(k) ~ zeta(2) * n * (log(n) + 2*gamma - 1 + zeta'(2)/zeta(2)), where gamma is Euler's constant (A001620). - _Amiram Eldar_, Dec 25 2024
%e A318491 1, 5/2, 7/3, 17/4, 11/5, 35/6, 15/7, 49/8, 34/9, 11/2, 23/11, 119/12, 27/13, 75/14, 77/15, 129/16, ...
%p A318491 f:= proc(n) local d;
%p A318491    numer(add(numtheory:-sigma(d)/d, d = numtheory:-divisors(n))) end proc:
%p A318491 map(f, [$1..65]); # _Robert Israel_, Jan 13 2025
%t A318491 Numerator[Table[Sum[DivisorSigma[-1, d], {d, Divisors[n]}], {n, 65}]]
%t A318491 Numerator[Table[Sum[DivisorSigma[1, d]/d, {d, Divisors[n]}], {n, 65}]]
%t A318491 Numerator[Table[Sum[d DivisorSigma[0, d], {d, Divisors[n]}]/n, {n, 65}]]
%t A318491 nmax = 65; Rest[Numerator[CoefficientList[Series[Sum[DivisorSigma[1, k] x^k/(k (1 - x^k)), {k, 1, nmax}], {x, 0, nmax}], x]]]
%t A318491 nmax = 65; Rest[Numerator[CoefficientList[Series[-Log[Product[(1 - x^k)^DivisorSigma[0, k], {k, 1, nmax}]], {x, 0, nmax}], x]]]
%o A318491 (PARI) a(n) = numerator(sumdiv(n, d, sumdiv(d, j, 1/j))); \\ _Michel Marcus_, Aug 28 2018
%Y A318491 Cf. A000005, A000203, A006171, A007429, A017665, A017666, A060640, A318492 (denominators).
%Y A318491 Cf. A001620, A013661, A306016.
%K A318491 nonn,frac,look
%O A318491 1,2
%A A318491 _Ilya Gutkovskiy_, Aug 27 2018