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A318317 Numerators of rational valued sequence whose Dirichlet convolution with itself yields A173557.

%I A318317 #17 May 11 2025 04:46:35
%S A318317 1,1,1,3,2,1,3,5,1,1,5,3,6,3,2,35,8,1,9,3,3,5,11,5,0,3,1,9,14,1,15,63,
%T A318317 5,4,6,3,18,9,6,5,20,3,21,15,1,11,23,35,-3,0,8,9,26,1,10,15,9,7,29,3,
%U A318317 30,15,3,231,12,5,33,3,11,3,35,5,36,9,0,27,15,3,39,35,3,10,41,9,16,21,14,25,44,1,18,33,15,23,18,63,48,-3,5,0,50,4,51,15,6
%N A318317 Numerators of rational valued sequence whose Dirichlet convolution with itself yields A173557.
%H A318317 Antti Karttunen, <a href="/A318317/b318317.txt">Table of n, a(n) for n = 1..65537</a>
%H A318317 Vaclav Kotesovec, <a href="/A318317/a318317.jpg">Graph - the asymptotic ratio (100000 terms)</a>
%F A318317 a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A173557(n) - Sum_{d|n, d>1, d<n} f(d) * f(n/d)) for n > 1.
%F A318317 From _Vaclav Kotesovec_, May 10 2025: (Start)
%F A318317 Let f(s) = Product_{p prime} (1 - 2/(p + p^s)).
%F A318317 Sum_{k=1..n} A318317(k) / A318318(k) ~ n^2 * sqrt(f(2)/(4*Pi*log(n))) * (1 + (1 - gamma - f'(2)/f(2) + 6*zeta'(2)/Pi^2) / (4*log(n))), where
%F A318317 f(2) = A307868 = Product_{p prime} (1 - 2/(p*(p+1))) = 0.471680613612997868...
%F A318317 f'(2)/f(2) = Sum_{p prime} 2*p*log(p) / ((p+1)*(p^2+p-2)) = 0.7254208328519472161058521308839896283514823... and gamma is the Euler-Mascheroni constant A001620. (End)
%t A318317 f[1] = 1; f[n_] := f[n] = 1/2 (Module[{fac = FactorInteger[n]}, If[n == 1, 1, Product[fac[[i, 1]] - 1, {i, Length[fac]}]]] - Sum[f[d]*f[n/d], {d, Divisors[n][[2 ;; -2]]}]); Table[Numerator[f[n]], {n, 1, 100}] (* _Vaclav Kotesovec_, May 10 2025 *)
%o A318317 (PARI)
%o A318317 up_to = 16384;
%o A318317 A173557(n) = my(f=factor(n)[, 1]); prod(k=1, #f, f[k]-1); \\ From A173557
%o A318317 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 A318317 v318317_18 = DirSqrt(vector(up_to, n, A173557(n)));
%o A318317 A318317(n) = numerator(v318317_18[n]);
%Y A318317 Cf. A173557, A318318 (denominators).
%Y A318317 Cf. also A317925, A317935.
%K A318317 sign,frac
%O A318317 1,4
%A A318317 _Antti Karttunen_, Aug 24 2018