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A318671 Numerators of the sequence whose Dirichlet convolution with itself yields A049599, number of (1+e)-divisors of n.

%I A318671 #6 Sep 03 2018 23:02:19
%S A318671 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,1,1,
%T A318671 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,11,1,1,1,1,
%U A318671 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,-3
%N A318671 Numerators of the sequence whose Dirichlet convolution with itself yields A049599, number of (1+e)-divisors of n.
%H A318671 Antti Karttunen, <a href="/A318671/b318671.txt">Table of n, a(n) for n = 1..65537</a>
%F A318671 a(n) = numerator 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.
%o A318671 (PARI)
%o A318671 up_to = (2^16)+1;
%o A318671 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 A318671 A049599(n) = factorback(apply(e -> (1+numdiv(e)),factor(n)[,2]));
%o A318671 v318671_72 = DirSqrt(vector(up_to, n, A049599(n)));
%o A318671 A318671(n) = numerator(v318671_72[n]);
%Y A318671 Cf. A049599, A318672 (denominators).
%K A318671 sign,frac
%O A318671 1,64
%A A318671 _Antti Karttunen_, Sep 03 2018