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A318454 Denominators of the sequence whose Dirichlet convolution with itself yields A001227, number of odd divisors of n.

%I A318454 #13 May 09 2025 10:49:30
%S A318454 1,2,1,8,1,2,1,16,1,2,1,8,1,2,1,128,1,2,1,8,1,2,1,16,1,2,1,8,1,2,1,
%T A318454 256,1,2,1,8,1,2,1,16,1,2,1,8,1,2,1,128,1,2,1,8,1,2,1,16,1,2,1,8,1,2,
%U A318454 1,1024,1,2,1,8,1,2,1,16,1,2,1,8,1,2,1,128,1,2,1,8,1,2,1,16,1,2,1,8,1,2,1,256,1,2,1,8,1,2,1,16,1
%N A318454 Denominators of the sequence whose Dirichlet convolution with itself yields A001227, number of odd divisors of n.
%C A318454 The sequence seems to give the denominators of several other similarly constructed "Dirichlet Square Roots".
%H A318454 Antti Karttunen, <a href="/A318454/b318454.txt">Table of n, a(n) for n = 1..16384</a>
%F A318454 a(n) = denominator of f(n), where f(1) = 1, f(n) = (1/2) * (A001227(n) - Sum_{d|n, d>1, d<n} f(d) * f(n/d)) for n > 1.
%F A318454 a(n) = 2^A318455(n).
%F A318454 Sum_{k=1..n} A318453(k) / a(k) ~ n/sqrt(2). - _Vaclav Kotesovec_, May 09 2025
%t A318454 f[1] = 1; f[n_] := f[n] = 1/2 (Sum[Mod[d, 2], {d, Divisors[n]}] - Sum[f[d] f[n/d], {d, Divisors[n][[2 ;; -2]]}]);
%t A318454 Table[f[n] // Denominator, {n, 1, 105}] (* _Jean-François Alcover_, Sep 13 2018 *)
%o A318454 (PARI)
%o A318454 up_to = 16384;
%o A318454 A001227(n) = numdiv(n>>valuation(n, 2)); \\ From A001227
%o A318454 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 A318454 v318453_54 = DirSqrt(vector(up_to, n, A001227(n)));
%o A318454 A318454(n) = denominator(v318453_54[n]);
%Y A318454 Cf. A001227.
%Y A318454 Cf. A318453 (numerators), A318455.
%K A318454 nonn,frac
%O A318454 1,2
%A A318454 _Antti Karttunen_ and _Andrew Howroyd_, Aug 29 2018