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A318498 Denominators of the sequence whose Dirichlet convolution with itself yields A061389, number of (1+phi)-divisors of n.

%I A318498 #8 Aug 30 2018 22:42:05
%S A318498 1,1,1,2,1,1,1,1,2,1,1,2,1,1,1,8,1,2,1,2,1,1,1,1,2,1,1,2,1,1,1,8,1,1,
%T A318498 1,4,1,1,1,1,1,1,1,2,2,1,1,8,2,2,1,2,1,1,1,1,1,1,1,2,1,1,2,16,1,1,1,2,
%U A318498 1,1,1,2,1,1,2,2,1,1,1,8,8,1,1,2,1,1,1,1,1,2,1,2,1,1,1,8,1,2,2,4,1,1,1,1,1
%N A318498 Denominators of the sequence whose Dirichlet convolution with itself yields A061389, number of (1+phi)-divisors of n.
%C A318498 The sequence seems to give the denominators of a few other similarly constructed rational valued sequences obtained as "Dirichlet Square Roots" (of possibly A092520 and A293443).
%H A318498 Antti Karttunen, <a href="/A318498/b318498.txt">Table of n, a(n) for n = 1..16384</a>
%F A318498 a(n) = denominator 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.
%F A318498 a(n) = 2^A318499(n).
%o A318498 (PARI)
%o A318498 up_to = 65537;
%o A318498 A061389(n) = factorback(apply(e -> (1+eulerphi(e)),factor(n)[,2]));
%o A318498 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 A318498 v318497_98 = DirSqrt(vector(up_to, n, A061389(n)));
%o A318498 A318497(n) = numerator(v318497_98[n]);
%o A318498 A318498(n) = denominator(v318497_98[n]);
%Y A318498 Cf. A061389, A318497 (numerators), A318499.
%Y A318498 Cf. also A299150, A046644.
%K A318498 nonn,frac
%O A318498 1,4
%A A318498 _Antti Karttunen_, Aug 30 2018