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A318497 Numerators of the sequence whose Dirichlet convolution with itself yields A061389, number of (1+phi)-divisors of n.

%I A318497 #10 Jul 29 2019 09:58:03
%S A318497 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,13,1,1,
%T A318497 1,1,1,1,1,1,1,1,1,1,1,1,1,3,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,-13,1,1,1,
%U A318497 1,1,1,1,1,1,1,1,1,1,1,1,3,3,1,1,1,1,1,1,1,1,1,1,1,1,1,1,13,1,1,1,1,1,1,1,1,1
%N A318497 Numerators of the sequence whose Dirichlet convolution with itself yields A061389, number of (1+phi)-divisors of n.
%C A318497 No zeros among the first 2^20 terms. This is most probably multiplicative, like A318498.
%H A318497 Antti Karttunen, <a href="/A318497/b318497.txt">Table of n, a(n) for n = 1..65537</a>
%F A318497 a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A061389(n) - Sum_{d|n, d>1, d<n} f(d) * f(n/d)) for n > 1.
%o A318497 (PARI)
%o A318497 up_to = 65537;
%o A318497 A061389(n) = factorback(apply(e -> (1+eulerphi(e)),factor(n)[,2]));
%o A318497 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 A318497 v318497_98 = DirSqrt(vector(up_to, n, A061389(n)));
%o A318497 A318497(n) = numerator(v318497_98[n]);
%Y A318497 Cf. A061389, A318314 (denominators).
%K A318497 sign,frac
%O A318497 1,16
%A A318497 _Antti Karttunen_, Aug 30 2018