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A318323 Numerators of rational valued sequence whose Dirichlet convolution with itself yields A046523, smallest number with same prime signature as n.

%I A318323 #7 Aug 24 2018 22:13:33
%S A318323 1,1,1,3,1,2,1,5,3,2,1,5,1,2,2,35,1,5,1,5,2,2,1,4,3,2,5,5,1,9,1,63,2,
%T A318323 2,2,35,1,2,2,4,1,9,1,5,5,2,1,55,3,5,2,5,1,4,2,4,2,2,1,9,1,2,5,231,2,
%U A318323 9,1,5,2,9,1,43,1,2,5,5,2,9,1,55,35,2,1,9,2,2,2,4,1,9,2,5,2,2,2,49,1,5,5,35,1,9,1,4,9
%N A318323 Numerators of rational valued sequence whose Dirichlet convolution with itself yields A046523, smallest number with same prime signature as n.
%C A318323 The first 2^20 terms are positive.
%H A318323 Antti Karttunen, <a href="/A318323/b318323.txt">Table of n, a(n) for n = 1..16384</a>
%F A318323 a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A046523(n) - Sum_{d|n, d>1, d<n} f(d) * f(n/d)) for n > 1.
%o A318323 (PARI)
%o A318323 up_to = 16384;
%o A318323 A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
%o A318323 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 A318323 v318323_24 = DirSqrt(vector(up_to, n, A046523(n)));
%o A318323 A318323(n) = numerator(v318323_24[n]);
%Y A318323 Cf. A046523, A318324 (gives the denominators).
%K A318323 nonn,frac
%O A318323 1,4
%A A318323 _Antti Karttunen_, Aug 24 2018