A365792 -id:A365792 - cp's OEIS frontend

A365791 a(n) = number of k <= b(n) such that rad(k) = rad(b(n)), where rad(n) = A007947(n) and b(n) = A126706(n).

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, 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, 2, 14, 2, 7, 5, 2, 3, 3, 2, 7, 5, 2, 3, 3, 9, 5, 2, 2, 4

Offset: 1

Michael De Vlieger, Sep 21 2023

nonn

Alternatively, position of A126706(n) in the list k*{R(k)} containing m such that A007947(m) = k, where k = A007947(n).
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.
Then k*{R(k)} is the list of numbers beginning with k, followed by nonsquarefree k*m such that rad(k*m) = k.
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.

			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.
a(2) = 10 since rad(b(2)) = rad(18) = 6, and 18 is the 3rd term in k*{R(6)}.
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.
a(4) = 4 since rad(b(4)) = rad(24) = 6, and 24 is the 4th term in k*{R(6)}.
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.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A007947, A126706, A365783, A365792.

nn = 270;
f[x_] := f[x] = Times @@ FactorInteger[x][[All, 1]];
t = Select[Range[nn], Nor[PrimePowerQ[#], SquareFreeQ[#]] &];
s = Map[f, t];
Map[Function[k, Set[r[k], k*Select[Range[nn/k], Divisible[k, f[#]] &]]], Union@ s];
Array[FirstPosition[r[s[[#]]], t[[#]] ][[1]] &, Length[t] ]

a(n) = A008479(A126706(n)).

a(n) > 1 for all n.

A365793 a(n) = number of k <= b(n) such that rad(k) = rad(b(n)), where rad(n) = A007947(n) and b(n) = A286708(n).

Original entry on oeis.org

5, 8, 6, 10, 11, 6, 8, 14, 5, 15, 16, 8, 11, 18, 5, 7, 12, 20, 21, 8, 7, 11, 14, 23, 18, 9, 24, 15, 6, 9, 25, 8, 5, 26, 8, 9, 13, 8, 6, 14, 18, 29, 19, 26, 11, 30, 19, 12, 8, 31, 10, 20, 32, 6, 32, 11, 16, 10, 33, 5, 10, 17, 22, 6, 8, 8, 13, 35, 28, 36, 8, 14

Offset: 1

Michael De Vlieger, Sep 22 2023

nonn

Alternatively, position of A126706(n) in the list k*{R(k)} containing m such that A007947(m) = k, where k = A007947(n).

			a(1) = 5 since rad(b(1)) = rad(36) = 6, and in the sequence k*{R(6)} = 6*{A003586} = {6, 12, 18, 24, 36, ...}, 36 is the 5th term.
a(2) = 8 since rad(b(2)) = rad(72) = 6, and 72 is the 8th term in k*{R(6)}.
a(3) = 6 since rad(b(3)) = rad(100) = 10, and in the sequence k*{R(10)} = 10*{A003592} = {10, 20, 40, 50, 80, 100, ...}, 100 is the 6th term, etc.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A007947, A286708, A365786, A365792.

nn = 4000;
f[x_] := f[x] = Times @@ FactorInteger[x][[All, 1]];
t = Select[
  Select[Range[nn], Nor[PrimePowerQ[#], SquareFreeQ[#]] &],
  AllTrue[FactorInteger[#][[All, -1]], # > 1 &] &];
s = Map[f, t];
Map[Function[k, Set[r[k], k*Select[Range[nn/k], Divisible[k, f[#]] &]]], Union@ s];
Array[FirstPosition[r[s[[#]]], t[[#]]][[1]] &, Length[t]]

a(n) = A008479(A286708(n)).

a(n) > 1 for all n.

cp's OEIS Frontend

A365791 a(n) = number of k <= b(n) such that rad(k) = rad(b(n)), where rad(n) = A007947(n) and b(n) = A126706(n).

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