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A318324 Denominators of rational valued sequence whose Dirichlet convolution with itself yields A046523, smallest number with same prime signature as n.

%I A318324 #5 Aug 24 2018 22:13:27
%S A318324 1,1,1,2,1,1,1,2,2,1,1,2,1,1,1,8,1,2,1,2,1,1,1,1,2,1,2,2,1,1,1,8,1,1,
%T A318324 1,4,1,1,1,1,1,1,1,2,2,1,1,8,2,2,1,2,1,1,1,1,1,1,1,1,1,1,2,16,1,1,1,2,
%U A318324 1,1,1,4,1,1,2,2,1,1,1,8,8,1,1,1,1,1,1,1,1,1,1,2,1,1,1,4,1,2,2,4,1,1,1,1,1
%N A318324 Denominators of rational valued sequence whose Dirichlet convolution with itself yields A046523, smallest number with same prime signature as n.
%H A318324 Antti Karttunen, <a href="/A318324/b318324.txt">Table of n, a(n) for n = 1..16384</a>
%F A318324 a(n) = denominator 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 A318324 (PARI)
%o A318324 up_to = 16384;
%o A318324 A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
%o A318324 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 A318324 v318323_24 = DirSqrt(vector(up_to, n, A046523(n)));
%o A318324 A318324(n) = denominator(v318323_24[n]);
%Y A318324 Cf. A046523, A318323 (numerators).
%K A318324 nonn,frac
%O A318324 1,4
%A A318324 _Antti Karttunen_, Aug 24 2018