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A317936 Numerators of sequence whose Dirichlet convolution with itself yields A100995 + A063524, that is, the characteristic function of A000961 (prime powers).

%I A317936 #7 Aug 23 2018 21:01:54
%S A317936 1,1,1,7,1,-1,1,17,7,-1,1,-5,1,-1,-1,139,1,-5,1,-5,-1,-1,1,-5,7,-1,17,
%T A317936 -5,1,3,1,263,-1,-1,-1,-31,1,-1,-1,-5,1,3,1,-5,-5,-1,1,19,7,-5,-1,-5,
%U A317936 1,-5,-1,-5,-1,-1,1,9,1,-1,-5,995,-1,3,1,-5,-1,3,1,-53,1,-1,-5,-5,-1,3,1,19,139,-1,1,9,-1,-1,-1,-5,1,9
%N A317936 Numerators of sequence whose Dirichlet convolution with itself yields A100995 + A063524, that is, the characteristic function of A000961 (prime powers).
%H A317936 Antti Karttunen, <a href="/A317936/b317936.txt">Table of n, a(n) for n = 1..65537</a>
%F A317936 a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A100995(n) - Sum_{d|n, d>1, d<n} f(d) * f(n/d)) for n > 1.
%o A317936 (PARI)
%o A317936 up_to = 65537;
%o A317936 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 A317936 v317936aux = DirSqrt(vector(up_to, n, if(1==n,1,isprimepower(n))));
%o A317936 A317936(n) = numerator(v317936aux[n]);
%o A317936 for(n=1,up_to,write("b317936.txt", n, " ", A317936(n)));
%Y A317936 Cf. A000961, A100995, A046644 (denominators).
%Y A317936 Cf. also A317939.
%K A317936 sign,frac
%O A317936 1,4
%A A317936 _Antti Karttunen_, Aug 14 2018