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A318672 Denominators of the sequence whose Dirichlet convolution with itself yields A049599, number of (1+e)-divisors of n.

%I A318672 #6 Sep 03 2018 23:02:27
%S A318672 1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,2,1,1,1,1,1,1,1,
%T A318672 1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1,1,1,1,2,1,2,1,1,1,1,1,1,1,8,1,1,1,1,
%U A318672 1,1,1,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,1
%N A318672 Denominators of the sequence whose Dirichlet convolution with itself yields A049599, number of (1+e)-divisors of n.
%C A318672 The sequence seems to give the denominators of a few other similarly constructed rational valued sequences obtained as "Dirichlet Square Roots" (possibly of A282446 and A318469).
%H A318672 Antti Karttunen, <a href="/A318672/b318672.txt">Table of n, a(n) for n = 1..16385</a>
%F A318672 a(n) = denominator of f(n), where f(1) = 1, f(n) = (1/2) * (A049599(n) - Sum_{d|n, d>1, d<n} f(d) * f(n/d)) for n > 1.
%F A318672 a(n) = 2^A318673(n).
%o A318672 (PARI)
%o A318672 up_to = (2^16)+1;
%o A318672 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 A318672 A049599(n) = factorback(apply(e -> (1+numdiv(e)),factor(n)[,2]));
%o A318672 v318671_62 = DirSqrt(vector(up_to, n, A049599(n)));
%o A318672 A318671(n) = numerator(v318671_62[n]);
%o A318672 A318672(n) = denominator(v318671_62[n]);
%o A318672 A318673(n) = valuation(A318672(n),2);
%Y A318672 Cf. A049599, A318671 (numerators), A318673.
%Y A318672 Cf. also A046644, A299150, A317932, A317934, A318498.
%K A318672 nonn,frac
%O A318672 1,8
%A A318672 _Antti Karttunen_, Sep 03 2018