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A318669 Numerators of the sequence whose Dirichlet convolution with itself yields A065769 ("Prime cascade").

%I A318669 #11 Sep 03 2018 23:02:05
%S A318669 1,1,1,7,3,1,5,25,5,3,7,7,11,5,3,363,13,5,17,21,5,7,19,25,51,11,13,35,
%T A318669 23,3,29,1335,7,13,15,35,31,17,11,75,37,5,41,49,15,19,43,363,115,51,
%U A318669 13,77,47,13,21,125,17,23,53,21,59,29,25,9923,33,7,61,91,19,15,67,125,71,31,51,119,35,11,73,1089,139,37,79,35,39,41,23
%N A318669 Numerators of the sequence whose Dirichlet convolution with itself yields A065769 ("Prime cascade").
%C A318669 Multiplicative because A065769 and A317932 are.
%H A318669 Antti Karttunen, <a href="/A318669/b318669.txt">Table of n, a(n) for n = 1..65537</a>
%F A318669 a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A065769(n) - Sum_{d|n, d>1, d<n} f(d) * f(n/d)) for n > 1.
%o A318669 (PARI)
%o A318669 up_to = 1+(2^16);
%o A318669 A003557(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 2] = max(0,f[i, 2]-1)); factorback(f); };
%o A318669 A065769(n) = { my(f=factor(n>>valuation(n,2))[, 1]~); (A003557(n) * factorback(vector(#f,i,precprime(f[i]-1)))); }; \\ _Antti Karttunen_, Sep 03 2018
%o A318669 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};
%o A318669 v318669_aux = DirSqrt(vector(up_to, n, A065769(n)));
%o A318669 A318669(n) = numerator(v318669_aux[n]);
%Y A318669 Cf. A065769, A317932 (denominators).
%K A318669 nonn,frac,mult
%O A318669 1,4
%A A318669 _Antti Karttunen_, Sep 03 2018