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A318650 Numerators of the sequence whose Dirichlet convolution with itself yields A057521, the powerful part of n.

%I A318650 #18 May 11 2025 04:56:45
%S A318650 1,1,1,15,1,1,1,49,35,1,1,15,1,1,1,603,1,35,1,15,1,1,1,49,99,1,181,15,
%T A318650 1,1,1,2023,1,1,1,525,1,1,1,49,1,1,1,15,35,1,1,603,195,99,1,15,1,181,
%U A318650 1,49,1,1,1,15,1,1,35,14875,1,1,1,15,1,1,1,1715,1,1,99,15,1,1,1,603,3235,1,1,15,1,1,1,49,1,35,1,15,1,1,1,2023,1
%N A318650 Numerators of the sequence whose Dirichlet convolution with itself yields A057521, the powerful part of n.
%C A318650 Multiplicative because A046644 and A057521 are.
%H A318650 Antti Karttunen, <a href="/A318650/b318650.txt">Table of n, a(n) for n = 1..65537</a>
%H A318650 Vaclav Kotesovec, <a href="/A318650/a318650_1.jpg">Graph - the asymptotic ratio (10^8 terms)</a>
%F A318650 a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A057521(n) - Sum_{d|n, d>1, d<n} f(d) * f(n/d)) for n > 1.
%F A318650 From _Vaclav Kotesovec_, May 10 2025, simplified May 11 2025: (Start)
%F A318650 Let f(s) = Product_{p prime} (1 - 1/p^(3*s-2) + 1/p^(3*s-3) + 1/p^s).
%F A318650 Sum_{k=1..n} A318650(k) / A046644(k) ~ n^(3/2) * sqrt(2*f(3/2)/(9*Pi*log(n))) * (1 + (2/3 - gamma - f'(3/2)/(2*f(3/2))) / (2*log(n))), where
%F A318650 f(3/2) = Product_{p prime} (1 + 2/p^(3/2) - 1/p^(5/2)) = A328013 = 3.51955505841710664719752940369857817...
%F A318650 f'(3/2)/f(3/2) = Sum_{p prime} (4*p - 3) * log(p) / (1 - 2*p - p^(5/2)) = -3.90914718020692131140714384422938370058563543737256496...
%F A318650 and gamma is the Euler-Mascheroni constant A001620. (End)
%t A318650 ff[p_, e_] := If[e > 1, p^e, 1]; a[1] = 1; a[n_] := Times @@ ff @@@ FactorInteger[n]; f[1] = 1; f[n_] := f[n] = 1/2 (a[n] - Sum[f[d] f[n/d], {d, Divisors[n][[2 ;; -2]]}]); Table[Numerator[f[n]], {n, 1, 100}] (* _Vaclav Kotesovec_, May 11 2025 *)
%o A318650 (PARI)
%o A318650 up_to = 65537;
%o A318650 A057521(n) = { my(f=factor(n)); prod(i=1, #f~, if(f[i, 2]>1, f[i, 1]^f[i, 2], 1)); }; \\ From A057521
%o A318650 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};
%o A318650 v318650_aux = DirSqrt(vector(up_to, n, A057521(n)));
%o A318650 A318650(n) = numerator(v318650_aux[n]);
%Y A318650 Cf. A057521, A046644 (denominators).
%Y A318650 Cf. also A317935, A318511, A318649.
%K A318650 nonn,frac,mult
%O A318650 1,4
%A A318650 _Antti Karttunen_, Aug 31 2018