cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A365784 a(n) = A126706(n) divided by its squarefree kernel.

Original entry on oeis.org

2, 3, 2, 4, 2, 6, 4, 2, 3, 8, 5, 2, 9, 4, 2, 3, 2, 12, 5, 2, 8, 2, 4, 3, 2, 16, 7, 3, 10, 4, 18, 8, 2, 3, 4, 2, 3, 2, 9, 4, 2, 24, 7, 2, 5, 4, 3, 2, 16, 27, 2, 4, 3, 2, 5, 8, 6, 4, 2, 9, 32, 14, 3, 20, 2, 3, 8, 2, 36, 2, 16, 15, 2, 4, 3, 2, 8, 11, 2, 7, 4, 25
Offset: 1

Views

Author

Michael De Vlieger, Sep 19 2023

Keywords

Comments

Let b(n) = A126706(n) and let squarefree kernel rad(n) = A007947(n).
a(n) > 1, rad(a(n)) | rad(b(n)).

Examples

			a(1) = 2 since b(1)/rad(b(1)) = 12/6 = 2.
a(2) = 3 since b(2)/rad(b(2)) = 18/6 = 3.
a(3) = 2 since b(3)/rad(b(3)) = 20/10 = 2.
a(4) = 4 since b(4)/rad(b(4)) = 24/6 = 4.
a(5) = 2 since b(5)/rad(b(5)) = 28/14 = 2.
a(6) = 6 since b(6)/rad(b(6)) = 36/6 = 6, etc.

Links

Crossrefs

Cf. A007947, A120944, A126706, A365783.

Programs

  • Mathematica
    Map[#/(Times @@ FactorInteger[#][[All, 1]]) &, Select[Range[12, 212], Nor[PrimePowerQ[#], SquareFreeQ[#]] &] ]
  • PARI
    apply(x->(x/factorback(factorint(x)[, 1])), select(x->(!issquarefree(x) && !isprimepower(x)), [1..300])) \\ Michel Marcus, Sep 19 2023

Formula

a(n) = A126706(n)/A365783(n) = A126706(n)/A007947(A126706(n)).

A365785 a(n) = k such that A120944(k) is the squarefree kernel of A126706(n).

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 2, 6, 4, 1, 2, 7, 1, 3, 8, 5, 10, 1, 4, 12, 2, 14, 6, 8, 15, 1, 3, 9, 2, 7, 1, 3, 19, 13, 8, 20, 14, 22, 4, 10, 24, 1, 5, 25, 8, 12, 16, 27, 2, 1, 28, 14, 18, 30, 11, 6, 8, 15, 34, 5, 1, 3, 22, 2, 36, 23, 7, 38, 1, 39, 3, 4, 41, 19, 27, 43, 8
Offset: 1

Views

Author

Michael De Vlieger, Sep 19 2023

Keywords

Examples

			Let b(n) = A126706(n), c(n) = A120944(n), and squarefree kernel rad(n) = A007947(n).
a(1) = 1 since c(1) = rad(b(1)) = rad(12) = 6.
a(2) = 1 since c(1) = rad(b(2)) = rad(18) = 6.
a(3) = 2 since c(2) = rad(b(3)) = rad(20) = 10.
a(4) = 1 since c(1) = rad(b(4)) = rad(24) = 6.
a(5) = 3 since c(3) = rad(b(5)) = rad(28) = 14, etc.

Links

Crossrefs

Cf. A007947, A120944, A126706, A365783.

Programs

  • Mathematica
    nn = 240;
s = Select[Range[12, nn], Nor[PrimePowerQ[#], SquareFreeQ[#]] &];
t = Select[Range[nn/2], And[SquareFreeQ[#], CompositeQ[#]] &];
Map[FirstPosition[t, Times @@ FactorInteger[#][[All, 1]]][[1]] &, s]

Formula

A120944(a(n)) = A007947(A126706(n)) = A365783(n).

A365790 a(n) = number of k <= b(n) such that rad(k) | b(n), where rad(n) = A007947(n) and b(n) = A126706(n).

Original entry on oeis.org

8, 10, 8, 11, 8, 14, 11, 9, 8, 15, 12, 9, 16, 11, 26, 8, 10, 18, 9, 10, 14, 28, 11, 32, 10, 20, 13, 8, 15, 11, 21, 14, 10, 8, 36, 10, 33, 31, 12, 12, 27, 23, 10, 11, 41, 12, 8, 31, 18, 24, 11, 38, 8, 11, 8, 14, 44, 12, 11, 11, 25, 16, 36, 19, 33, 8, 14, 11, 26
Offset: 1

Views

Author

Michael De Vlieger, Sep 21 2023

Keywords

Comments

Alternatively, position of A126706(n) in the list R(rad(n)) of k such that rad(k) | n, where rad(n) = A007947(n). Note that rad(b(n)) < b(n) for all n.
Let prime p divide n. The set R(rad(n)) is a list of numbers beginning with the empty product 1 and including all k such that p | k implies p | rad(n). For example, R(6) = A003586. All k in A003586 are such that no prime q coprime to 6 divides k.

Examples

			a(1) = 8 since rad(b(1)) = rad(12) = 6, and in the sequence R(6) = A003586 = {1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, ...}, 12 is the 8th term.
a(2) = 10 since rad(b(2)) = rad(18) = 6, and 18 is the 10th term in R(6).
a(3) = 8 since rad(b(3)) = rad(20) = 10, and in the sequence R(10) = A003592 = {1, 2, 4, 5, 8, 10, 16, 20, ...}, 20 is the 8th term.
a(4) = 11 since rad(b(4)) = rad(24) = 6, and 24 is the 11th term in R(6).
a(5) = 8 since rad(b(5)) = rad(28) = 14, and in the sequence R(14) = A003591 = {1, 2, 4, 7, 8, 14, 16, 28, ...}, 28 is the 8th term, etc.

Links

Crossrefs

Cf. A007947, A010846, A126706, A365783, A365791.

Programs

  • Mathematica
    nn = 220;
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], Select[Range[nn], Divisible[k, f[#]] &]]], Union@ s];
Array[FirstPosition[r[s[[#]]], t[[#]] ][[1]] &, Length[t] ]

Formula

a(n) = A010846(A126706(n)).

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).

Original entry on oeis.org

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

Views

Author

Michael De Vlieger, Sep 21 2023

Keywords

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.

Examples

			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.

Links

Crossrefs

Cf. A007947, A126706, A365783, A365792.

Programs

  • 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] ]

Formula

a(n) = A008479(A126706(n)).
a(n) > 1 for all n.
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.