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A318453 Numerators of the sequence whose Dirichlet convolution with itself yields A001227, number of odd divisors of n.

%I A318453 #14 May 09 2025 10:47:52
%S A318453 1,1,1,3,1,1,1,5,1,1,1,3,1,1,1,35,1,1,1,3,1,1,1,5,1,1,1,3,1,1,1,63,1,
%T A318453 1,1,3,1,1,1,5,1,1,1,3,1,1,1,35,1,1,1,3,1,1,1,5,1,1,1,3,1,1,1,231,1,1,
%U A318453 1,3,1,1,1,5,1,1,1,3,1,1,1,35,1,1,1,3,1,1,1,5,1,1,1,3,1,1,1,63,1,1,1,3,1,1,1,5,1
%N A318453 Numerators of the sequence whose Dirichlet convolution with itself yields A001227, number of odd divisors of n.
%H A318453 Antti Karttunen, <a href="/A318453/b318453.txt">Table of n, a(n) for n = 1..65537</a>
%H A318453 Vaclav Kotesovec, <a href="/A318453/a318453.jpg">Graph - the asymptotic ratio (16384 terms)</a>
%F A318453 a(n) = numerator 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 A318453 Sum_{k=1..n} A318453(k) / A318454(k) ~ n/sqrt(2). - _Vaclav Kotesovec_, May 09 2025
%t A318453 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 A318453 Table[f[n] // Numerator, {n, 1, 105}] (* _Jean-François Alcover_, Sep 13 2018 *)
%o A318453 (PARI)
%o A318453 up_to = 16384;
%o A318453 A001227(n) = numdiv(n>>valuation(n, 2)); \\ From A001227
%o A318453 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 A318453 v318453_54 = DirSqrt(vector(up_to, n, A001227(n)));
%o A318453 A318453(n) = numerator(v318453_54[n]);
%o A318453 A318454(n) = denominator(v318453_54[n]);
%Y A318453 Cf. A001227.
%Y A318453 Cf. A318454 (gives the denominators).
%Y A318453 Differs from A318313 for the first time at n=81, where a(81) = 1, while A318313(81) = 3.
%K A318453 nonn,frac
%O A318453 1,4
%A A318453 _Antti Karttunen_ and _Andrew Howroyd_, Aug 29 2018