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 1-6 of 6 results.

A117283 Numbers k for which the cototient k-phi(k) is a cube.

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 12, 13, 14, 16, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 63, 67, 71, 73, 79, 81, 83, 89, 96, 97, 101, 103, 107, 109, 112, 113, 115, 124, 127, 128, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 187, 191, 193, 197, 199, 211, 223, 227
Offset: 1

Views

Author

Luc Stevens (lms022(AT)yahoo.com), Apr 23 2006

Keywords

Examples

			63 is in the sequence because 63-phi(63) = 27, which is a cube.

Links

Crossrefs

Cf. A000010, A000578, A051953.
Cf. A117284, A117285, A117286, A117287, A117288, A117289.

Programs

  • Mathematica
    Select[Range[230], IntegerQ[(# - EulerPhi[ # ])^(1/3)] &] (* Stefan Steinerberger, Apr 24 2006 *)
  • PARI
    isok(k) = ispower(k-eulerphi(k), 3); \\ Michel Marcus, Mar 23 2021

A117284 Numbers k for which the cototient k-phi(k) is a triangular number.

Original entry on oeis.org

1, 2, 3, 5, 7, 9, 10, 11, 13, 17, 19, 23, 29, 31, 37, 39, 41, 43, 45, 47, 52, 53, 54, 55, 57, 59, 61, 67, 68, 71, 73, 79, 83, 85, 89, 90, 97, 101, 103, 107, 109, 113, 114, 117, 127, 129, 131, 137, 139, 149, 151, 157, 159, 163, 167, 168, 173, 175, 179, 181, 191, 193
Offset: 1

Views

Author

Luc Stevens (lms022(AT)yahoo.com), Apr 23 2006

Keywords

Examples

			39 is in the sequence because 39-phi(39) = 15, which is a triangular number.

Links

Crossrefs

Cf. A000010, A000217, A051953.
Cf. A117283, A117285, A117286, A117287, A117288, A117289.

Programs

  • Mathematica
    Select[Range[200],OddQ[Sqrt[8(#-EulerPhi[#])+1]]&] (* Harvey P. Dale, Jun 29 2015 *)
  • PARI
    isok(n) = ispolygonal(n - eulerphi(n), 3); \\ Michel Marcus, Feb 26 2014

Extensions

Offset corrected by Amiram Eldar, Mar 23 2021

A117285 Numbers k for which the cototient k-phi(k) is a pentagonal number.

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 13, 17, 18, 19, 20, 22, 23, 25, 29, 30, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 75, 79, 83, 89, 97, 101, 102, 103, 107, 109, 110, 113, 127, 131, 132, 137, 139, 140, 149, 151, 155, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 203, 211, 223
Offset: 1

Views

Author

Luc Stevens (lms022(AT)yahoo.com), Apr 23 2006

Keywords

Examples

			30 is in the sequence because 30-phi(30) = 22, which is a pentagonal number.

Links

Crossrefs

Cf. A000010, A000326, A051953.
Cf. A117283, A117284, A117286, A117287, A117288, A117289.

Programs

  • Mathematica
    pentQ[n_] := n == 0 || IntegerQ[(Sqrt[24*n + 1] + 1)/6]; Select[Range[250], pentQ[# - EulerPhi[#]] &] (* Amiram Eldar, Mar 23 2021 *)
  • PARI
    isok(n) = ispolygonal(n - eulerphi(n), 5); \\ Michel Marcus, Feb 26 2014

Extensions

Offset corrected by Amiram Eldar, Mar 23 2021

A117286 Numbers k for which the cototient k-phi(k) is a hexagonal number.

Original entry on oeis.org

1, 2, 3, 5, 7, 10, 11, 13, 17, 19, 23, 29, 31, 37, 39, 41, 43, 47, 52, 53, 55, 59, 61, 67, 71, 73, 79, 83, 89, 90, 97, 101, 103, 107, 109, 113, 117, 127, 129, 131, 137, 139, 149, 151, 157, 163, 167, 168, 173, 179, 181, 191, 193, 197, 199, 200, 205, 211, 223, 227, 229
Offset: 1

Views

Author

Luc Stevens (lms022(AT)yahoo.com), Apr 23 2006

Keywords

Comments

55 is in the sequence because 55-phi(55) = 15, which is a hexagonal number.

Links

Crossrefs

Cf. A000010, A000384, A051953.
Cf. A117283, A117284, A117285, A117287, A117288, A117289.

Programs

  • Mathematica
    cthQ[n_]:=Module[{c=n-EulerPhi[n]},IntegerQ[(Sqrt[1+8c]+1)/4]]; Join[{1}, Select[Range[250],cthQ]] (* Harvey P. Dale, Jan 16 2012 *)
  • PARI
    isok(n) = ispolygonal(n - eulerphi(n), 6); \\ Michel Marcus, Feb 26 2014

Extensions

Corrected by T. D. Noe, Oct 25 2006

A117287 Numbers k for which the cototient k-phi(k) is a heptagonal number.

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 13, 15, 17, 19, 23, 29, 31, 34, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 159, 163, 167, 173, 175, 179, 181, 189, 191, 193, 196, 197, 199, 208, 211, 223, 227, 229, 233
Offset: 1

Views

Author

Luc Stevens (lms022(AT)yahoo.com), Apr 23 2006

Keywords

Examples

			49 is in the sequence because 49-phi(49) = 7, which is a heptagonal number.

Links

Crossrefs

Cf. A000010, A000566, A051953.
Cf. A117283, A117284, A117285, A117286, A117288, A117289.

Programs

  • Mathematica
    hnQ[n_]:=Module[{c=n-EulerPhi[n]},IntegerQ[(3+Sqrt[9+40c])/10]]; Join[{1},Select[Range[0,250],hnQ]] (* Harvey P. Dale, Mar 26 2011 *)
  • PARI
    isok(n) = ispolygonal(n - eulerphi(n), 7); \\ Michel Marcus, Feb 26 2014

Extensions

Offset corrected by Amiram Eldar, Mar 23 2021

A117288 Numbers k for which the cototient k-phi(k) is an octagonal number.

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 12, 13, 14, 16, 17, 19, 23, 29, 31, 37, 41, 43, 45, 47, 53, 57, 59, 61, 67, 71, 73, 76, 79, 83, 85, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 144, 149, 151, 157, 160, 163, 167, 173, 176, 179, 181, 184, 188, 191, 193, 197, 199, 211, 223
Offset: 1

Views

Author

Luc Stevens (lms022(AT)yahoo.com), Apr 23 2006

Keywords

Examples

			45 is in the sequence because 45-phi(45) = 21, which is an octagonal number.

Links

Crossrefs

Cf. A000010, A000567, A051953.
Cf. A117283, A117284, A117285, A117286, A117287, A117289.

Programs

  • Mathematica
    octQ[n_] := n == 0 || IntegerQ[(Sqrt[48*n + 16] + 4)/12]; Select[Range[250], octQ[# - EulerPhi[#]] &] (* Amiram Eldar, Mar 23 2021 *)
  • PARI
    isok(n) = ispolygonal(n - eulerphi(n), 8); \\ Michel Marcus, Feb 26 2014

Extensions

Offset corrected by Amiram Eldar, Mar 23 2021
Showing 1-6 of 6 results.

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