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A318318 Denominators of rational valued sequence whose Dirichlet convolution with itself yields A173557.

%I A318318 #9 May 10 2025 12:13:55
%S A318318 1,2,1,8,1,2,1,16,2,1,1,8,1,2,1,128,1,4,1,4,1,2,1,16,1,1,2,8,1,1,1,
%T A318318 256,1,1,1,16,1,2,1,8,1,2,1,8,1,2,1,128,2,1,1,4,1,4,1,16,1,1,1,4,1,2,
%U A318318 2,1024,1,2,1,1,1,1,1,32,1,1,1,8,1,1,1,64,8,1,1,8,1,2,1,16,1,2,1,8,1,2,1,256,1,4,2,1,1,1,1,8,1
%N A318318 Denominators of rational valued sequence whose Dirichlet convolution with itself yields A173557.
%C A318318 Not multiplicative because A318317 contains zeros.
%C A318318 Differs from A317926 at n = 200, 400, 600, 800, 900, 1200, 1400, 1600, 1800, 2200, 2400, 2700, 2800, 3200, 3600, 3800, 4050, 4200, 4400, 4600, 4800, 4900, 5200, ..., which seem to be a subsequence of positions of zeros in A318317.
%C A318318 Here a(200) = 1, while A317926(200) = 2.
%H A318318 Antti Karttunen, <a href="/A318318/b318318.txt">Table of n, a(n) for n = 1..65537</a>
%F A318318 a(n) = denominator 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.
%t A318318 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[Denominator[f[n]], {n, 1, 100}] (* _Vaclav Kotesovec_, May 10 2025 *)
%o A318318 (PARI)
%o A318318 up_to = 16384;
%o A318318 A173557(n) = my(f=factor(n)[, 1]); prod(k=1, #f, f[k]-1); \\ From A173557
%o A318318 DirSqrt(v) = {my(n=#v, u=vector(nA173557)); 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 A318318 v318317_18 = DirSqrt(vector(up_to, n, A173557(n)));
%o A318318 A318318(n) = denominator(v318317_18[n]);
%Y A318318 Cf. A173557, A318317 (numerators).
%Y A318318 Cf. also A317926.
%K A318318 nonn,frac
%O A318318 1,2
%A A318318 _Antti Karttunen_, Aug 24 2018