cp's OEIS Frontend

A317831 Numerators of rational valued sequence f whose Dirichlet convolution with itself yields A000203 (sigma, the sum of divisors).

%I A317831 #14 May 09 2025 12:14:28
%S A317831 1,3,2,19,3,3,4,63,9,9,6,19,7,6,6,867,9,27,10,57,8,9,12,63,11,21,11,
%T A317831 19,15,9,16,3069,12,27,12,171,19,15,14,189,21,12,22,57,27,18,24,867,
%U A317831 41,33,18,133,27,33,18,63,20,45,30,57,31,24,18,22199,21,18,34,171,24,18,36,567,37,57,22,95,24,21,40,2601,227,63,42,19,27,33
%N A317831 Numerators of rational valued sequence f whose Dirichlet convolution with itself yields A000203 (sigma, the sum of divisors).
%H A317831 Antti Karttunen, <a href="/A317831/b317831.txt">Table of n, a(n) for n = 1..16384</a>
%H A317831 Vaclav Kotesovec, <a href="/A317831/a317831.jpg">Graph - the asymptotic ratio (10000 terms)</a>
%H A317831 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>
%F A317831 a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A000203(n) - Sum_{d|n, d>1, d<n} f(d) * f(n/d)) for n > 1.
%F A317831 Sum_{k=1..n} A317831(k) / A317832(k) ~ n^2 * sqrt(Pi/(24*log(n))) * (1 - (gamma - 1 + 6*zeta'(2)/Pi^2) / (4*log(n))), where gamma is the Euler-Mascheroni constant A001620. - _Vaclav Kotesovec_, May 09 2025
%o A317831 (PARI)
%o A317831 A317831perA317832(n) = if(1==n,n,(sigma(n)-sumdiv(n,d,if((d>1)&&(d<n),A317831perA317832(d)*A317831perA317832(n/d),0)))/2);
%o A317831 A317831(n) = numerator(A317831perA317832(n));
%o A317831 (PARI) for(n=1, 100, print1(numerator(direuler(p=2, n, 1/((1-p*X)*(1-X))^(1/2))[n]), ", ")) \\ _Vaclav Kotesovec_, May 09 2025
%Y A317831 Cf. A317832 (gives the denominators).
%Y A317831 Cf. also A000203, A299151.
%K A317831 nonn,frac
%O A317831 1,2
%A A317831 _Antti Karttunen_, Aug 10 2018