cp's OEIS Frontend

A317930 Numerators of rational valued sequence whose Dirichlet convolution with itself yields A234840, which is a multiplicative permutation of natural numbers.

%I A317930 #7 Aug 24 2018 22:11:51
%S A317930 1,3,1,27,19,3,61,135,3,57,11,27,281,183,19,2835,101,9,5,513,61,33,
%T A317930 263,135,1083,843,5,1647,29,57,59,15309,11,303,1159,81,1811,15,281,
%U A317930 2565,1091,183,157,297,57,789,409,2835,11163,3249,101,7587,541,15,209,8235,5,87,31,513,7,177,183,168399,5339,33,1013,2727
%N A317930 Numerators of rational valued sequence whose Dirichlet convolution with itself yields A234840, which is a multiplicative permutation of natural numbers.
%C A317930 Multiplicative because A234840 is.
%C A317930 Question: Are all terms positive? No negative terms in range 1 .. 2^17. Also (checked for n <= 2^17) the denominators seem to be given by A317932.
%H A317930 Antti Karttunen, <a href="/A317930/b317930.txt">Table of n, a(n) for n = 1..16384</a>
%F A317930 a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A234840(n) - Sum_{d|n, d>1, d<n} f(d) * f(n/d)) for n > 1.
%o A317930 (PARI)
%o A317930 up_to = 16384;
%o A317930 A234840(n) = if(n<=1,n,my(f = factor(n)); for(i=1, #f~, if(2==f[i,1], f[i,1]++, if(3==f[i,1], f[i,1]--, f[i,1] = prime(-1+A234840(1+primepi(f[i,1])))))); factorback(f)); \\ _Antti Karttunen_, Aug 23 2018
%o A317930 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 A317930 v317930aux = DirSqrt(vector(up_to, n, A234840(n)));
%o A317930 A317930(n) = numerator(v317930aux[n]);
%Y A317930 Cf. A234840, A317932 (seems to give denominators, see A261179).
%Y A317930 Cf. also A317929.
%K A317930 nonn,frac,mult
%O A317930 1,2
%A A317930 _Antti Karttunen_, Aug 23 2018