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A318658 Denominators of the sequence whose Dirichlet convolution with itself yields A087003, a(2n) = 0 and a(2n+1) = moebius(2n+1).

%I A318658 #15 Sep 23 2018 21:31:23
%S A318658 1,1,2,1,2,1,2,1,8,1,2,1,2,1,4,1,2,1,2,1,4,1,2,1,8,1,16,1,2,1,2,1,4,1,
%T A318658 4,1,2,1,4,1,2,1,2,1,16,1,2,1,8,1,4,1,2,1,4,1,4,1,2,1,2,1,16,1,4,1,2,
%U A318658 1,4,1,2,1,2,1,16,1,4,1,2,1,128,1,2,1,4,1,4,1,2,1,4,1,4,1,4,1,2,1,16,1,2,1,2,1,8
%N A318658 Denominators of the sequence whose Dirichlet convolution with itself yields A087003, a(2n) = 0 and a(2n+1) = moebius(2n+1).
%C A318658 The sequence seems to give the denominators of several other similarly constructed "Dirichlet Square Roots".
%H A318658 Antti Karttunen, <a href="/A318658/b318658.txt">Table of n, a(n) for n = 1..16384</a>
%F A318658 a(n) = denominator of f(n), where f(1) = 1, f(n) = (1/2) * (A087003(n) - Sum_{d|n, d>1, d<n} f(d) * f(n/d)) for n > 1.
%F A318658 a(n) = 2^A318659(n).
%F A318658 a(2n) = 1, a(2n-1) = A046644(2n-1) = A318512(2n-1), for all n >= 1.
%o A318658 (PARI)
%o A318658 up_to = 65537;
%o A318658 A087003(n) = ((n%2)*moebius(n)); \\ I.e. a(n) = A000035(n)*A008683(n).
%o A318658 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 A318658 v318657_18 = DirSqrt(vector(up_to, n, A087003(n)));
%o A318658 A318657(n) = numerator(v318657_18[n]);
%o A318658 A318658(n) = denominator(v318657_18[n]);
%Y A318658 Cf. A005187, A087003, A318657 (numerators), A318659.
%K A318658 nonn,frac
%O A318658 1,3
%A A318658 _Antti Karttunen_, Aug 31 2018