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A318664 Numerators of the sequence whose Dirichlet convolution with itself yields A064664, the inverse permutation of EKG-sequence.

%I A318664 #12 Sep 01 2018 22:28:19
%S A318664 1,1,5,1,5,-1,7,3,-1,-1,10,3,14,-1,-7,5,33,59,37,9,-10,-1,43,-1,-1,-1,
%T A318664 181,13,57,89,61,15,-29,-1,-45,31,67,-1,-41,1,37,129,81,11,301,-1,89,
%U A318664 21,1,26,-97,10,50,-93,-47,-5,-109,-1,107,-33,115,-1,411,15,-43,201,64,33,-127,56,67,181,69,-1,283,35,-31,255,151,7
%N A318664 Numerators of the sequence whose Dirichlet convolution with itself yields A064664, the inverse permutation of EKG-sequence.
%H A318664 Antti Karttunen, <a href="/A318664/b318664.txt">Table of n, a(n) for n = 1..16384</a>
%H A318664 <a href="/index/Ed#EKG">Index entries for sequences related to EKG sequence</a>
%F A318664 a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A064664(n) - Sum_{d|n, d>1, d<n} f(d) * f(n/d)) for n > 1.
%F A318664 For n >= 2, a(2*A000040(n)) = -1.
%o A318664 (PARI)
%o A318664 v064413 = readvec("b064413_upto65539_terms_only.txt"); \\ From b-file of A064413 prepared beforehand.
%o A318664 A064413(n) = v064413[n];
%o A318664 m064664 = Map();
%o A318664 for(n=1,65539,mapput(m064664,A064413(n),n));
%o A318664 A064664(n) = mapget(m064664,n);
%o A318664 up_to = (2^14);
%o A318664 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 A318664 v318664_65 = DirSqrt(vector(up_to, n, A064664(n)));
%o A318664 A318664(n) = numerator(v318664_65[n]);
%o A318664 A318665(n) = denominator(v318664_65[n]);
%Y A318664 Cf. A064664, A304526, A304527, A305293, A305294, A318665 (denominators).
%Y A318664 Cf. also A317929, A317930.
%K A318664 sign,frac
%O A318664 1,3
%A A318664 _Antti Karttunen_, Sep 01 2018