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 1-7 of 7 results.

A077092 Fixed points of iteration in A077091.

Original entry on oeis.org

0, 4, 24, 8064, 34944, 35520, 38880, 69480, 268560, 420096, 1054944, 2946560, 3054080, 5660160, 6621120, 9768960, 10264320, 25885760, 29062656, 33933312, 36484992, 38707200, 78532608, 163418112, 260601600, 458987520, 4044349440
Offset: 1

Views

Author

Labos Elemer, Oct 31 2002

Keywords

Comments

When iteration of f(k) = phi(sigma(k)-phi(k)) is started at various initial values, not ending in cycles and converging, it ends at these fixed points.
a(28) <= 11435212800. a(29) <= 15083274240. a(30) <= 90215424000. - Donovan Johnson, Dec 14 2009

Examples

			n=30: FixedPointList={30,32,46,20,16,22,12,8,10,6,4},end=4; n=94: FixedPointList={94,42,24},end=24. n=41708: FixedPointList={41708,26064,32352,21216,15232,8064},end=8064; n=12100: FixedPointList={12100,24000,34944},end=34944.

Crossrefs

Cf. A077090, A077091, A077093, A077094, A077095, A077096, A077099, A077100.

Programs

  • Mathematica
    f[x_] := EulerPhi[DivisorSigma[1, x]-EulerPhi[x]] Do[s=NestList[f, n, 100]; s1=Part[s, 99]; s2=Part[s, 100]; If[Equal[s1, s2]&&!PrimeQ[n], Print[{n, s1}]], {n, 1, 1000}]

Extensions

a(9) corrected and a(11)-a(27) from Donovan Johnson, Dec 14 2009

A077094 Numbers k such that iterating phi(sigma(k)-phi(k)) starting from k leads to the fixed point 4.

Original entry on oeis.org

4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 25, 26, 27, 28, 30, 32, 33, 34, 35, 38, 39, 44, 45, 46, 49, 51, 54, 55, 56, 57, 58, 62, 63, 65, 68, 70, 77, 85, 87, 91, 93, 95, 99, 104, 111, 119, 121, 129, 134, 143, 145, 153, 158, 161, 169, 189, 205, 209, 215, 221, 245
Offset: 1

Views

Author

Labos Elemer, Oct 31 2002

Keywords

Comments

Probably this sequence is finite, with 92 terms of which the last is 6241.

Examples

			n=6241: FixedPointList={6241,104,54,32,20,16,22,12,8,10,6,4}, end=4.

Links

Crossrefs

Cf. A000010 (phi), A000203 (sigma).
Cf. A077090, A077091, A077092, A077093, A077095, A077096.

Programs

  • Mathematica
    f[x_] := EulerPhi[DivisorSigma[1, x]-EulerPhi[x]]; Do[s=NestList[f, n, 100]; s1=Part[s, 99]; s2=Part[s, 100]; If[Equal[s1, s2]&&Equal[s1, 4], Print[{n, s1}]], {n, 1, 1000000}]
f4[n_]:=FixedPoint[EulerPhi[DivisorSigma[1,#]-EulerPhi[#]]&,n,50]==4; Select[Range[250],f4] (* Harvey P. Dale, May 01 2021 *)

Extensions

Definition corrected by Harvey P. Dale, May 01 2021

A077091 Composites c, such that when iteration of f(k) = phi(sigma(k)-phi(k)) is started at c it ends at a fixed point > 1.

Original entry on oeis.org

4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 38, 39, 42, 44, 45, 46, 49, 51, 54, 55, 56, 57, 58, 62, 63, 65, 68, 69, 70, 74, 75, 77, 82, 85, 86, 87, 91, 93, 94, 95, 99, 104, 111, 115, 119, 121, 125, 129, 133, 134, 143, 145, 153
Offset: 1

Views

Author

Labos Elemer, Oct 31 2002

Keywords

Examples

			n=30: FixedPointList={30,32,46,20,16,22,12,8,10,6,4},end=4; n=94:FixedPointList={94,42,24},end=24.

Crossrefs

Cf. A077090, A077092, A077093, A077094, A077095, A077096.

Programs

  • Mathematica
    f[x_] := EulerPhi[DivisorSigma[1, x]-EulerPhi[x]] Do[s=NestList[f, n, 100]; s1=Part[s, 99]; s2=Part[s, 100]; If[Equal[s1, s2]&&!PrimeQ[n], Print[{n, s1}]], {n, 1, 1000}]

Extensions

1 removed by Sean A. Irvine, May 05 2025

A077093 Smallest initial values leading to fixed points listed in A077092.

Original entry on oeis.org

1, 4, 24, 7530, 12100, 32784, 34950, 69480, 121104, 420096, 1018980, 2846484, 2946560, 4160040, 5387484, 5512800, 5729520, 13108800, 23524452, 24799080, 25885760, 31382952, 53344272, 100929036, 150057300, 352636452
Offset: 1

Views

Author

Labos Elemer, Oct 31 2002

Keywords

Examples

			n=12100: FixedPointList={12100,24000,34944},end=34944; n=121104: FixedPointList={121104,268560}, end=268560.

Crossrefs

Cf. A077090, A077091, A077092, A077094, A077095, A077096, A077099, A077100.

Programs

  • Mathematica
    f[x_] := EulerPhi[DivisorSigma[1, x]-EulerPhi[x]]; Do[s=NestList[f, n, 100]; s1=Part[s, 99]; s2=Part[s, 100]; If[Equal[s1, s2]&&!PrimeQ[n], Print[{n, s1}]], {n, 1, 1000}] (*n=site if fixed point appears; s1=fixed point*)

Extensions

a(11)-a(26) from Sean A. Irvine, May 06 2025

A077095 Numbers k such that iterating phi(sigma(k)-phi(k)) starting from k leads to the fixed point 24.

Original entry on oeis.org

24, 42, 69, 74, 75, 82, 86, 94, 115, 125, 133, 155, 185, 187, 203, 289, 299, 323, 341, 361, 377, 437, 1681
Offset: 1

Views

Author

Labos Elemer, Oct 31 2002

Keywords

Comments

Probably this sequence is finite, with 23 terms of which the last is 1681.

Examples

			n=1641: FixedPointList={1681,82,42,24}, end=24.

Crossrefs

Cf. A000010 (phi), A000203 (sigma).
Cf. A077090, A077091, A077092, A077093, A077094, A077096.

Programs

  • Mathematica
    f[x_] := EulerPhi[DivisorSigma[1, x]-EulerPhi[x]]; Do[s=NestList[f, n, 100]; s1=Part[s, 99]; s2=Part[s, 100]; If[Equal[s1, s2]&&Equal[s1, 4], Print[{n, s1}]], {n, 1, 1000000}]
fp24Q[n_]:=FixedPoint[EulerPhi[DivisorSigma[1,#]-EulerPhi[#]]&,n,20]==24; Select[ Range[1700],fp24Q] (* Harvey P. Dale, Mar 12 2023 *)

A077096 Numbers k such that iterating phi(sigma(k)-phi(k)) starting from k leads to the fixed point 8064.

Original entry on oeis.org

7530, 8064, 9678, 9828, 9990, 10002, 10290, 10464, 11000, 11004, 11172, 11350, 11510, 11572, 11814, 11930, 12006, 12192, 12348, 12472, 12636, 12654, 12726, 12750, 12772, 12972, 13332, 13372, 13420, 13440, 13626, 13648, 13656, 13695
Offset: 1

Views

Author

Labos Elemer, Oct 31 2002

Keywords

Examples

			n=41354: FixedPointList={41354,13440,15232,8064}, end=8064.

Crossrefs

Cf. A077090, A077091, A077092, A077093, A077094, A077095.

Programs

  • Mathematica
    f[x_] := EulerPhi[DivisorSigma[1, x]-EulerPhi[x]] Do[s=NestList[f, n, 100]; s1=Part[s, 99]; s2=Part[s, 100]; If[Equal[s1, s2]&&Equal[s1, 8064], Print[n]], {n, 1, 1000000}]
fp[n_]:=FixedPoint[EulerPhi[DivisorSigma[1,#]-EulerPhi[#]]&,n,100]==8064; Select[ Range[14000],fp] (* Harvey P. Dale, May 05 2013 *)

A077088 a(n) = phi(sigma(n) - phi(n)), where phi is Euler's totient function and sigma is the sum of divisors function, with a(1) = 0.

Original entry on oeis.org

0, 1, 1, 4, 1, 4, 1, 10, 6, 6, 1, 8, 1, 6, 8, 22, 1, 20, 1, 16, 8, 12, 1, 24, 10, 8, 10, 20, 1, 32, 1, 46, 12, 18, 8, 78, 1, 12, 16, 36, 1, 24, 1, 32, 18, 20, 1, 36, 8, 72, 16, 36, 1, 32, 16, 32, 20, 30, 1, 72, 1, 20, 32, 72, 12, 60, 1, 46, 24, 32, 1, 108, 1, 24, 24, 48, 12, 48, 1, 60
Offset: 1

Views

Author

Labos Elemer, Nov 04 2002

Keywords

Comments

a(p) = 1 for p prime. Otherwise a(n) is even.

Examples

			a(10) = 6 because sigma(10) = 18 and phi(10) = 4, and so phi(18 - 4) = phi(14) = 6.
a(11) = 1 because sigma(11) = 12 and phi(11) = 10, so phi(12 - 10) = phi(2) = 1.
a(12) = 8 because sigma(12) = 28 and phi(12) = 4, so phi(28 - 4) = phi(24) = 8.

Links

Crossrefs

Cf. A000010, A000203, A051612, A065387. See iterations in A077090-A077100.

Programs

Formula

a(1) = 0; and for n > 1, a(n) = A000010(A051612(n)).

Extensions

Value of a(1) clarified by Antti Karttunen, Mar 04 2018
Showing 1-7 of 7 results.

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