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.

A365792 a(n) = number of k <= b(n) such that rad(k) | b(n), where rad(n) = A007947(n) and b(n) = A286708(n).

Original entry on oeis.org

14, 18, 15, 21, 23, 16, 19, 26, 13, 29, 30, 20, 23, 32, 14, 18, 24, 35, 36, 18, 19, 24, 28, 39, 83, 21, 40, 29, 15, 20, 42, 21, 13, 43, 18, 22, 27, 21, 15, 28, 33, 46, 91, 104, 25, 47, 34, 23, 22, 50, 24, 36, 51, 16, 120, 26, 32, 24, 52, 13, 22, 33, 39, 16, 19
Offset: 1

Views

Author

Michael De Vlieger, Sep 22 2023

Keywords

Comments

Alternatively, position of A286708(n) in the list R(rad(n)) of k such that rad(k) | n, where rad(n) = A007947(n).

Examples

			a(1) = 14 since rad(b(1)) = rad(36) = 6, and in the sequence R(6) = A003586 = {1, 2, 3, 4, 6, 8, 9, ..., 36, ...}, 36 is the 14th term.
a(2) = 18 since rad(b(2)) = rad(72) = 6, and 72 is the 18th term in R(6).
a(3) = 15 since rad(b(3)) = rad(100) = 10, and in the sequence R(10) = A003592 = {1, 2, 4, 5, 8, 10, ..., 100, ...}, 100 is the 15th term, etc.

Links

Crossrefs

Cf. A007947, A010846, A286708, A365786, A365793.

Programs

  • Mathematica
    nn = 3300; 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], Select[Range[nn], Divisible[k, f[#]] &]]], Union@ s];
Array[FirstPosition[r[s[[#]]], t[[#]]][[1]] &, Length[t]]

A365787 a(n) = A286708(n) divided by its squarefree kernel.

Original entry on oeis.org

6, 12, 10, 18, 24, 14, 20, 36, 15, 48, 54, 28, 40, 72, 21, 22, 50, 96, 108, 45, 26, 56, 80, 144, 30, 44, 162, 100, 33, 75, 192, 34, 35, 216, 63, 52, 98, 38, 39, 112, 160, 288, 42, 60, 88, 324, 200, 135, 46, 384, 68, 250, 432, 51, 90, 104, 196, 76, 486, 55, 147
Offset: 1

Views

Author

Michael De Vlieger, Sep 19 2023

Keywords

Comments

Permutation of numbers that are not prime powers A024619.

Examples

			a(1) = 2 since b(1)/rad(b(1)) = 36/6 = 6.
a(2) = 3 since b(2)/rad(b(2)) = 72/6 = 12.
a(3) = 2 since b(3)/rad(b(3)) = 100/10 = 10.
a(4) = 4 since b(4)/rad(b(4)) = 108/6 = 18.
a(5) = 2 since b(5)/rad(b(5)) = 144/6 = 24.
a(6) = 6 since b(6)/rad(b(6)) = 196/14 = 14, etc

Links

Crossrefs

Cf. A007947, A024619, A120944, A286708, A365786.

Programs

  • Mathematica
    nn = 5000;
s = Rest@ Select[Union@ Flatten@
  Table[a^2*b^3, {b, nn^(1/3)}, {a, Sqrt[nn/b^3]}],
  Not @* PrimePowerQ];
t = Select[Range[nn/6], And[SquareFreeQ[#], CompositeQ[#]] &];
Map[FirstPosition[t, Times @@ FactorInteger[#][[All, 1]]][[1]] &, s]

Formula

a(n) = A286708(n)/A007947(A286708(n)) = A286708(n)/A365786(n).
Let b(n) = A286708(n) and let squarefree kernel rad(n) = A007947(n). a(n) >= n such that rad(a(n)) | 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

Views

Author

Michael De Vlieger, Sep 22 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).

Examples

			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.

Links

Crossrefs

Cf. A007947, A286708, A365786, A365792.

Programs

  • Mathematica
    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]]

Formula

a(n) = A008479(A286708(n)).
a(n) > 1 for all n.

A365789 Position of A365787(n) in A024619.

Original entry on oeis.org

1, 3, 2, 6, 10, 4, 7, 17, 5, 25, 29, 12, 20, 42, 8, 9, 26, 61, 69, 23, 11, 31, 48, 96, 13, 22, 111, 64, 14, 44, 134, 15, 16, 154, 36, 28, 62, 18, 19, 72, 109, 210, 21, 34, 54, 240, 139, 89, 24, 288, 39, 181, 329, 27, 55, 66, 137, 45, 374, 30, 99, 161, 236, 32
Offset: 1

Views

Author

Michael De Vlieger, Sep 19 2023

Keywords

Comments

A365787(n) = A286708(n)/A007947(A286708(n)).
Permutation of natural numbers.

Examples

			Let b(n) = A286708(n), rad(n) = A007947(n), and c(n) = A024619(n).
a(1) = 1 since b(1)/rad(b(1)) = 36/6 = 6 = c(1).
a(2) = 3 since b(2)/rad(b(2)) = 72/6 = 12 = c(3).
a(3) = 2 since b(3)/rad(b(3)) = 100/10 = 10 = c(2).
a(4) = 6 since b(4)/rad(b(4)) = 108/6 = 18 = c(6).
a(5) = 10 since b(5)/rad(b(5)) = 144/6 = 24 = c(10).
a(6) = 4 since b(6)/rad(b(6)) = 196/14 = 14 = c(4), etc.

Links

Crossrefs

Cf. A007947, A024619, A120944, A286708, A365786, A365787.

Programs

  • Mathematica
    nn = 3600;
s = Rest@
  Select[Union@ Flatten@
   Table[a^2*b^3, {b, nn^(1/3)}, {a, Sqrt[nn/b^3]}],
   Not @* PrimePowerQ];
t = Select[Range[2, nn], Not @* PrimePowerQ];
Map[FirstPosition[t, #/(Times @@ FactorInteger[#][[All, 1]])][[1]] &, s]
Showing 1-4 of 4 results.

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