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A317939 Numerators of sequence whose Dirichlet convolution with itself yields A080339 = A010051 (characteristic function of primes) + A063524 (1, 0, 0, 0, ...).

%I A317939 #7 Aug 23 2018 21:01:43
%S A317939 1,1,1,-1,1,-1,1,1,-1,-1,1,3,1,-1,-1,-5,1,3,1,3,-1,-1,1,-5,-1,-1,1,3,
%T A317939 1,3,1,7,-1,-1,-1,-15,1,-1,-1,-5,1,3,1,3,3,-1,1,35,-1,3,-1,3,1,-5,-1,
%U A317939 -5,-1,-1,1,-15,1,-1,3,-21,-1,3,1,3,-1,3,1,35,1,-1,3,3,-1,3,1,35,-5,-1,1,-15,-1,-1,-1,-5,1,-15,-1,3,-1,-1,-1,-63,1,3,3
%N A317939 Numerators of sequence whose Dirichlet convolution with itself yields A080339 = A010051 (characteristic function of primes) + A063524 (1, 0, 0, 0, ...).
%H A317939 Antti Karttunen, <a href="/A317939/b317939.txt">Table of n, a(n) for n = 1..65537</a>
%F A317939 a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A010051(n) - Sum_{d|n, d>1, d<n} f(d) * f(n/d)) for n > 1.
%o A317939 (PARI)
%o A317939 up_to = 65537;
%o A317939 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 A317939 v317939aux = DirSqrt(vector(up_to, n, if(1==n,1,isprime(n))));
%o A317939 A317939(n) = numerator(v317939aux[n]);
%Y A317939 Cf. A010051, A080339, A046644 (denominators).
%Y A317939 Cf. also A257098, A317830, A317936, A317937.
%K A317939 sign,frac
%O A317939 1,12
%A A317939 _Antti Karttunen_, Aug 14 2018