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%I #5 Oct 06 2023 11:10:16
%S 2,3,2,4,2,5,3,2,2,6,4,2,7,3,2,2,2,8,3,2,5,2,3,3,2,9,4,2,6,3,10,5,2,2,
%T 4,2,3,2,4,3,2,11,3,2,5,3,2,2,7,12,2,4,2,2,2,4,6,3,2,4,13,6,3,8,2,2,4,
%U 2,14,2,7,5,2,3,3,2,7,5,2,3,3,9,5,2,2,4
%N a(n) = number of k <= b(n) such that rad(k) = rad(b(n)), where rad(n) = A007947(n) and b(n) = A126706(n).
%C Alternatively, position of A126706(n) in the list k*{R(k)} containing m such that A007947(m) = k, where k = A007947(n).
%C The set R(k) is a list of numbers beginning with the empty product 1 and including all m such that p | m implies p | n. For example, R(6) = A003586. All k in A003586 are such that no prime q coprime to 6 divides k.
%C Then k*{R(k)} is the list of numbers beginning with k, followed by nonsquarefree k*m such that rad(k*m) = k.
%C The number k is composite and the only squarefree term in k*{R(k)} and appears in A120944; the rest of the list is in A126706.
%H Michael De Vlieger, <a href="/A365791/b365791.txt">Table of n, a(n) for n = 1..10000</a>
%F a(n) = A008479(A126706(n)).
%F a(n) > 1 for all n.
%e a(1) = 2 since rad(b(1)) = rad(12) = 6, and in the sequence k*{R(6)} = 6*{A003586} = {6, 12, 18, 24, 36, ...}, 12 is the 2nd term.
%e a(2) = 10 since rad(b(2)) = rad(18) = 6, and 18 is the 3rd term in k*{R(6)}.
%e a(3) = 2 since rad(b(3)) = rad(20) = 10, and in the sequence k*{R(10)} = 10*{A003592} = {10, 20, 40, 50, 80, ...}, 20 is the 2nd term.
%e a(4) = 4 since rad(b(4)) = rad(24) = 6, and 24 is the 4th term in k*{R(6)}.
%e a(5) = 2 since rad(b(5)) = rad(28) = 14, and in the sequence k*{R(14)} = 14*{A003591} = {14, 28, 56, 98, 112, ...}, 28 is the 2nd term, etc.
%t nn = 270;
%t f[x_] := f[x] = Times @@ FactorInteger[x][[All, 1]];
%t t = Select[Range[nn], Nor[PrimePowerQ[#], SquareFreeQ[#]] &];
%t s = Map[f, t];
%t Map[Function[k, Set[r[k], k*Select[Range[nn/k], Divisible[k, f[#]] &]]], Union@ s];
%t Array[FirstPosition[r[s[[#]]], t[[#]] ][[1]] &, Length[t] ]
%Y Cf. A007947, A126706, A365783, A365792.
%K nonn
%O 1,1
%A _Michael De Vlieger_, Sep 21 2023