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A318667 Numerators of the sequence whose Dirichlet convolution with itself yields A318307, which is multiplicative with A318307(p^e) = 2^A002487(e).

%I A318667 #6 Sep 03 2018 23:01:55
%S A318667 1,1,1,1,1,1,1,3,1,1,1,1,1,1,1,-5,1,1,1,1,1,1,1,3,1,1,3,1,1,1,1,31,1,
%T A318667 1,1,1,1,1,1,3,1,1,1,1,1,1,1,-5,1,1,1,1,1,3,1,3,1,1,1,1,1,1,1,-43,1,1,
%U A318667 1,1,1,1,1,3,1,1,1,1,1,1,1,-5,-5,1,1,1,1,1,1,3,1,1,1,1,1,1,1,31,1,1,1,1,1,1,1,3,1
%N A318667 Numerators of the sequence whose Dirichlet convolution with itself yields A318307, which is multiplicative with A318307(p^e) = 2^A002487(e).
%C A318667 Multiplicative because A318307 and A317934 are.
%H A318667 Antti Karttunen, <a href="/A318667/b318667.txt">Table of n, a(n) for n = 1..65537</a>
%F A318667 a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A318307(n) - Sum_{d|n, d>1, d<n} f(d) * f(n/d)) for n > 1.
%o A318667 (PARI)
%o A318667 up_to = 1+(2^16);
%o A318667 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 A318667 A002487(n) = { my(a=1, b=0); while(n>0, if(bitand(n, 1), b+=a, a+=b); n>>=1); (b); }; \\ From A002487
%o A318667 A318307(n) = factorback(apply(e -> 2^A002487(e),factor(n)[,2]));
%o A318667 v318667_aux = DirSqrt(vector(up_to, n, A318307(n)));
%o A318667 A318667(n) = numerator(v318667_aux[n]);
%Y A318667 Cf. A318307, A317934 (denominators).
%K A318667 sign,frac,mult
%O A318667 1,8
%A A318667 _Antti Karttunen_, Sep 03 2018