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A318319 Numerators of rational valued sequence whose Dirichlet convolution with itself yields A064989.

%I A318319 #7 Aug 24 2018 22:13:09
%S A318319 1,1,1,3,3,1,5,5,3,3,7,3,11,5,3,35,13,3,17,9,5,7,19,5,27,11,5,15,23,3,
%T A318319 29,63,7,13,15,9,31,17,11,15,37,5,41,21,9,19,43,35,75,27,13,33,47,5,
%U A318319 21,25,17,23,53,9,59,29,15,231,33,7,61,39,19,15,67,15,71,31,27,51,35,11,73,105,35,37,79,15,39,41,23,35,83,9,55,57
%N A318319 Numerators of rational valued sequence whose Dirichlet convolution with itself yields A064989.
%C A318319 Multiplicative because A064989 is.
%C A318319 No negative terms among the first 2^20 terms.
%H A318319 Antti Karttunen, <a href="/A318319/b318319.txt">Table of n, a(n) for n = 1..16384</a>
%H A318319 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%F A318319 a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A064989(n) - Sum_{d|n, d>1, d<n} f(d) * f(n/d)) for n > 1.
%o A318319 (PARI)
%o A318319 up_to = 16384;
%o A318319 A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
%o A318319 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 A318319 v318319aux = DirSqrt(vector(up_to, n, A064989(n)));
%o A318319 A318319(n) = numerator(v318319aux[n]);
%Y A318319 Cf. A064989, A317932 (seems to give denominators, see A261179).
%Y A318319 Cf. also A318321.
%K A318319 nonn,frac,mult
%O A318319 1,4
%A A318319 _Antti Karttunen_, Aug 24 2018