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A317940 Numerators of sequence whose Dirichlet convolution with itself yields A046644.

%I A317940 #8 Aug 24 2018 22:12:29
%S A317940 1,1,1,7,1,1,1,9,7,1,1,7,1,1,1,427,1,7,1,7,1,1,1,9,7,1,9,7,1,1,1,471,
%T A317940 1,1,1,49,1,1,1,9,1,1,1,7,7,1,1,427,7,7,1,7,1,9,1,9,1,1,1,7,1,1,7,
%U A317940 4099,1,1,1,7,1,1,1,63,1,1,7,7,1,1,1,427,427,1,1,7,1,1,1,9,1,7,1,7,1,1,1,471,1,7,7,49,1,1,1,9,1
%N A317940 Numerators of sequence whose Dirichlet convolution with itself yields A046644.
%C A317940 Multiplicative because A046644 is.
%C A317940 No negative terms among the first 2^20 terms. Is the sequence nonnegative?
%H A317940 Antti Karttunen, <a href="/A317940/b317940.txt">Table of n, a(n) for n = 1..65537</a>
%F A317940 a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A046644(n) - Sum_{d|n, d>1, d<n} f(d) * f(n/d)) for n > 1.
%o A317940 (PARI)
%o A317940 up_to = 65537;
%o A317940 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 A317940 A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
%o A317940 A046644(n) = factorback(apply(e -> 2^A005187(e),factor(n)[,2]));
%o A317940 v317940aux = DirSqrt(vector(up_to, n, A046644(n)));
%o A317940 A317940(n) = numerator(v317940aux[n]);
%Y A317940 Cf. A005187, A046644, A317934 (denominators), A317941.
%K A317940 nonn,frac,mult
%O A317940 1,4
%A A317940 _Antti Karttunen_, Aug 14 2018