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

1,1

Comments

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.

Links

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

Formula

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

a(n) > 1 for all n.

Example

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.

Mathematica

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] ]

Crossrefs

Cf. A007947, A126706, A365783, A365792.

Sequence in context: A111982 A323236 A319702 * A331252 A162908 A373558

Adjacent sequences: A365788 A365789 A365790 * A365792 A365793 A365794

Keyword

nonn

Author

Michael De Vlieger, Sep 21 2023

Status

approved