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A318444 Numerators of the sequence whose Dirichlet convolution with itself yields A057660(n) = Sum_{k=1..n} n/gcd(n,k).

%I A318444 #16 May 10 2025 04:30:49
%S A318444 1,3,7,35,21,21,43,239,195,63,111,245,157,129,147,6851,273,585,343,
%T A318444 735,301,333,507,1673,1643,471,3011,1505,813,441,931,50141,777,819,
%U A318444 903,6825,1333,1029,1099,5019,1641,903,1807,3885,4095,1521,2163,47957,6555,4929,1911,5495,2757,9033,2331,10277,2401,2439,3423
%N A318444 Numerators of the sequence whose Dirichlet convolution with itself yields A057660(n) = Sum_{k=1..n} n/gcd(n,k).
%C A318444 Because A057660 contains only odd values, A046644 gives the sequence of denominators. Because both of those sequences are multiplicative, this is also.
%C A318444 General formula: if k >= 0, m > 0, and the Dirichlet generating function is zeta(s-k)^m * f(s), where f(s) has all possible poles at points less than k+1, then Sum_{j=1..n} a(j) ~ n^(k+1) * log(n)^(m-1) * f(k+1) / ((k+1) * Gamma(m)) * (1 + (m-1)*(m*gamma - 1/(k+1) + f'(k+1)/f(k+1)) / log(n)), where gamma is the Euler-Mascheroni constant A001620 and Gamma() is the Gamma function. - _Vaclav Kotesovec_, May 10 2025
%H A318444 Antti Karttunen, <a href="/A318444/b318444.txt">Table of n, a(n) for n = 1..16384</a>
%H A318444 Vaclav Kotesovec, <a href="/A318444/a318444.jpg">Graph - the asymptotic ratio (10000 terms)</a>
%F A318444 a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A057660(n) - Sum_{d|n, d>1, d<n} f(d) * f(n/d)) for n > 1.
%F A318444 Sum_{k=1..n} A318444(k) / A046644(k) ~ n^3 * Pi^(-3/2) * sqrt(2*zeta(3)/(3*log(n))) * (1 + (1/3 - gamma/2 + 3*zeta'(2)/Pi^2 - zeta'(3)/(2*zeta(3))) / (2*log(n))), where gamma is the Euler-Mascheroni constant A001620. - _Vaclav Kotesovec_, May 10 2025
%t A318444 a57660[n_] := DivisorSigma[2, n^2]/DivisorSigma[1, n^2];
%t A318444 f[1] = 1; f[n_] := f[n] = 1/2 (a57660[n] - Sum[f[d]*f[n/d], {d, Divisors[ n][[2 ;; -2]]}]);
%t A318444 Table[f[n] // Numerator, {n, 1, 60}] (* _Jean-François Alcover_, Sep 13 2018 *)
%o A318444 (PARI)
%o A318444 up_to = 16384;
%o A318444 A057660(n) = sumdivmult(n, d, eulerphi(d)*d); \\ From A057660
%o A318444 DirSqrt(v) = {my(n=#v, u=vector(n)); u[1]=1; for(n=2, n, u[n]=(v[n]/v[1] - sumdiv(n, d, if(d>1&&d<n, u[d]*u[n/d], 0)))/2); u}; \\ From A317937.
%o A318444 v318444aux = DirSqrt(vector(up_to, n, A057660(n)));
%o A318444 A318444(n) = numerator(v318444aux[n]);
%o A318444 (PARI) for(n=1, 100, print1(numerator(direuler(p=2, n, ((1-p*X)/((1-p^2*X)*(1-X)))^(1/2))[n]), ", ")) \\ _Vaclav Kotesovec_, May 09 2025
%Y A318444 Cf. A057660, A046644 (denominators).
%Y A318444 Cf. also A318443.
%K A318444 nonn,frac,mult
%O A318444 1,2
%A A318444 _Antti Karttunen_ and _Andrew Howroyd_, Aug 29 2018