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A318313 Numerators of the sequence whose Dirichlet convolution with itself yields A068068, number of odd unitary divisors of n.

%I A318313 #15 May 10 2025 04:05:36
%S A318313 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 A318313 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 A318313 1,3,1,1,1,5,1,1,1,3,1,1,1,35,3,1,1,3,1,1,1,5,1,1,1,3,1,1,1,63,1,1,1,3,1,1,1,5,1
%N A318313 Numerators of the sequence whose Dirichlet convolution with itself yields A068068, number of odd unitary divisors of n.
%H A318313 Antti Karttunen, <a href="/A318313/b318313.txt">Table of n, a(n) for n = 1..65537</a>
%H A318313 Vaclav Kotesovec, <a href="/A318313/a318313.jpg">Graph - the asymptotic ratio (16384 terms)</a>
%F A318313 a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A068068(n) - Sum_{d|n, d>1, d<n} f(d) * f(n/d)) for n > 1.
%F A318313 Sum_{k=1..n} A318313(k) / A318314(k) ~ 2*n/Pi. - _Vaclav Kotesovec_, May 10 2025
%o A318313 (PARI)
%o A318313 up_to = 16384;
%o A318313 A068068(n) = (2^omega(n>>valuation(n, 2))); \\ From A068068
%o A318313 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 A318313 v318313_15 = DirSqrt(vector(up_to, n, A068068(n)));
%o A318313 A318313(n) = numerator(v318313_15[n]);
%Y A318313 Cf. A068068, A318314 (denominators).
%Y A318313 Differs from A318453 for the first time at n=81, where a(81) = 3, while A318453(81) = 1.
%K A318313 nonn,frac,mult
%O A318313 1,4
%A A318313 _Antti Karttunen_ and _Andrew Howroyd_, Aug 29 2018