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A317834 Numerators of rational valued sequence whose Dirichlet convolution with itself yields A078899 (the ordinal transform of A006530, the largest prime factor of n).

%I A317834 #10 Dec 19 2021 08:21:41
%S A317834 1,1,1,7,1,3,1,17,11,3,1,19,1,3,5,139,1,23,1,19,5,3,1,39,19,3,45,19,1,
%T A317834 13,1,263,5,3,9,77,1,3,5,55,1,13,1,19,43,3,1,387,27,47,5,19,1,59,9,71,
%U A317834 5,3,1,43,1,3,51,995,9,13,1,19,5,25,1,87,1,3,59,19,13,13,1,707,467,3,1,59,9,3,5,71,1,53,13,19,5,3,9,1069,1
%N A317834 Numerators of rational valued sequence whose Dirichlet convolution with itself yields A078899 (the ordinal transform of A006530, the largest prime factor of n).
%C A317834 The first negative term is a(216) = -97.
%H A317834 Antti Karttunen, <a href="/A317834/b317834.txt">Table of n, a(n) for n = 1..16384</a>
%F A317834 a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A078899(n) - Sum_{d|n, d>1, d<n} f(d) * f(n/d)) for n > 1.
%t A317834 gpf[n_] := If[n == 1, 1, FactorInteger[n][[-1, 1]]];
%t A317834 b[_] = 1;
%t A317834 A078899[n_] := A078899[n] = With[{t = gpf[n]}, b[t]++];
%t A317834 f[n_] := f[n] = If[n == 1, 1, (1/2)(A078899[n] -
%t A317834      Sum[If[1<d<n, f[d] f[n/d], 0], {d, Divisors[n]}])];
%t A317834 a[n_] := Numerator[f[n]];
%t A317834 Array[a, 100] (* _Jean-François Alcover_, Dec 19 2021 *)
%o A317834 (PARI)
%o A317834 up_to = 16384;
%o A317834 ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
%o A317834 A006530(n) = if(n>1, vecmax(factor(n)[, 1]), 1);
%o A317834 v078899 = ordinal_transform(vector(up_to,n,A006530(n)));
%o A317834 A078899(n) = v078899[n];
%o A317834 A317834aux(n) = if(1==n,n,(A078899(n)-sumdiv(n,d,if((d>1)&&(d<n),A317834aux(d)*A317834aux(n/d),0)))/2);
%o A317834 A317834(n) = numerator(A317834aux(n));
%Y A317834 Cf. A078899, A046644 (denominators).
%Y A317834 Cf. also A305799, A317833, A317830.
%K A317834 sign,frac
%O A317834 1,4
%A A317834 _Antti Karttunen_, Aug 12 2018