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A317929 Numerators of rational valued sequence whose Dirichlet convolution with itself yields A235199, which is a multiplicative permutation of natural numbers.

%I A317929 #6 Aug 24 2018 22:11:45
%S A317929 1,1,3,3,7,3,5,5,27,7,17,9,13,5,21,35,11,27,19,21,15,17,23,15,147,13,
%T A317929 135,15,43,21,59,63,51,11,35,81,37,19,39,35,41,15,29,51,189,23,73,105,
%U A317929 75,147,33,39,53,135,119,25,57,43,31,63,61,59,135,231,91,51,67,33,69,35,107,135,47,37,441,57,85,39
%N A317929 Numerators of rational valued sequence whose Dirichlet convolution with itself yields A235199, which is a multiplicative permutation of natural numbers.
%C A317929 Multiplicative because A235199 is.
%C A317929 Question: Are all terms positive? No negative terms in range 1 .. 2^18. Also checked up to n = 2^18 that the denominators match with A299150.
%H A317929 Antti Karttunen, <a href="/A317929/b317929.txt">Table of n, a(n) for n = 1..16384</a>
%F A317929 a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A235199(n) - Sum_{d|n, d>1, d<n} f(d) * f(n/d)) for n > 1.
%o A317929 (PARI)
%o A317929 up_to = 16384;
%o A317929 A235199(n) = if(n<=4,n,my(f = factor(n)); for(i=1, #f~, if(5==f[i,1], f[i,1] += 2, if(7==f[i,1], f[i,1] -= 2, my(k=primepi(f[i,1])); if(k>4, f[i,1] = prime(A235199(k)))))); factorback(f));
%o A317929 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 A317929 v317929aux = DirSqrt(vector(up_to, n, A235199(n)));
%o A317929 A317929(n) = numerator(v317929aux[n]);
%Y A317929 Cf. A235199, A299150 (seems to give the denominators).
%Y A317929 Cf. also A317930.
%K A317929 nonn,frac,mult
%O A317929 1,3
%A A317929 _Antti Karttunen_, Aug 23 2018