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A317937 Numerators of sequence whose Dirichlet convolution with itself yields sequence A001221 (omega n) + A063524 (1, 0, 0, 0, ...).

%I A317937 #10 Aug 26 2018 12:26:31
%S A317937 1,1,1,3,1,3,1,5,3,3,1,7,1,3,3,35,1,7,1,7,3,3,1,11,3,3,5,7,1,3,1,63,3,
%T A317937 3,3,9,1,3,3,11,1,3,1,7,7,3,1,75,3,7,3,7,1,11,3,11,3,3,1,1,1,3,7,231,
%U A317937 3,3,1,7,3,3,1,19,1,3,7,7,3,3,1,75,35,3,1,1,3,3,3,11,1,1,3,7,3,3,3,133,1,7,7,9,1,3,1,11,3
%N A317937 Numerators of sequence whose Dirichlet convolution with itself yields sequence A001221 (omega n) + A063524 (1, 0, 0, 0, ...).
%C A317937 The first negative term is a(210) = -7.
%H A317937 Antti Karttunen, <a href="/A317937/b317937.txt">Table of n, a(n) for n = 1..65537</a>
%F A317937 a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A001221(n) - Sum_{d|n, d>1, d<n} f(d) * f(n/d)) for n > 1.
%o A317937 (PARI)
%o A317937 A317937aux(n) = if(1==n,n,(omega(n)-sumdiv(n,d,if((d>1)&&(d<n),A317937aux(d)*A317937aux(n/d),0)))/2);
%o A317937 A317937(n) = numerator(A317937aux(n));
%o A317937 (PARI)
%o A317937 \\ DirSqrt(v) finds u such that v = v[1]*dirmul(u, u).
%o A317937 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 A317937 apply(numerator, DirSqrt(vector(100, n, if(1==n, 1, omega(n))))) \\ _Andrew Howroyd_, Aug 13 2018
%Y A317937 Cf. A001221, A063524, A046644 (denominators).
%Y A317937 Cf. also A317831, A317925, A317933, A317845, A317846, A317936, A317938, A317939.
%K A317937 sign,frac
%O A317937 1,4
%A A317937 _Antti Karttunen_, Aug 12 2018