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.

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

A077090 When iteration of f(x) = phi(sigma(x) - phi(x)) is started at initial values listed here it ends up in a cycle of length greater than 1.

Original entry on oeis.org

36, 40, 48, 50, 52, 60, 64, 66, 72, 76, 78, 80, 81, 84, 88, 90, 92, 96, 98, 100, 102, 105, 106, 108, 110, 112, 114, 116, 117, 118, 120, 122, 123, 124, 126, 128, 130, 132, 135, 136, 138, 140, 141, 142, 144, 146, 147, 148, 150, 152, 154, 156, 159, 160, 162, 164
Offset: 1

Views

Author

Labos Elemer, Oct 31 2002

Keywords

Comments

Below 2000 there are only these two cycles of length 3: 36, 78, 48; and 144, 280, 192.
For most composite numbers below 80, the iterated trajectory of f settles on a single-number cycle of 4; those numbers are not in this sequence. - Alonso del Arte, Nov 29 2013

Examples

			36 is in the sequence because f(36) = 78, f(78) = 48 and f(48) = 36, which is a cycle of length 3.
38 is not in the sequence because iterating f from 38 gives the trajectory 38, 12, 8, 10, 6, 4, 4, 4, ... where the cycle has a length of 1.

Links

Crossrefs

Cf. A077088, A077091, A077092, A077093, A077094, A077095, A077096.

Programs

  • Mathematica
    f[x_] := EulerPhi[DivisorSigma[1, x] - EulerPhi[x]]; Do[s = NestList[f, n, 100]; s1 = Part[s, 98]; s2 = Part[s, 99]; s3 = Part[s, 100]; If[ !Equal[s1, s2], k = k + 1; Print[{k, n, s1, s2, s3}]], {n, 2, 1000}]
f[n_] := EulerPhi[DivisorSigma[1, n]-EulerPhi[n]]; cycleQ[n_] := With[{nn = NestWhileList[f, n, Unequal, All]}, nn[[-1]] != nn[[-2]]]; Select[Range[200], cycleQ] (* Jean-François Alcover, Nov 29 2013 *)
  • PARI
    f(x)=if(x>35,eulerphi(sigma(x)-eulerphi(x)),1)
is(n)=my(t=f(n),h=f(t)); while(t!=h, h=f(f(h)); t=f(t)); f(t)!=t \\ Charles R Greathouse IV, Nov 29 2013

Extensions

Name corrected by Charles R Greathouse IV, Nov 29 2013

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

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 *)

A243995 Fixed points of the function a(n) = phi(sigma(n) - phi(n) - d(n)), where phi(n) is the Euler totient function, sigma(n) the sum of divisors of n and d(n) the number of divisors of n.

Original entry on oeis.org

1, 18, 96, 560, 720, 1056, 1760, 2000, 3264, 7392, 30960, 50800, 172080, 173440, 9561600, 13304064, 195522560, 1017856448, 1026021024, 1568010240, 2817884160
Offset: 1

Views

Author

Paolo P. Lava, Jun 18 2014

Keywords

Comments

a(22) > 10^10. - Amiram Eldar, Jan 27 2019

Examples

			sigma(96) = 252, phi(96) = 32, d(96) = 12 and phi(252 - 32 - 12) = 96.

Crossrefs

Cf. A000005, A000010, A000203, A077092.

Programs

  • Maple
    with(numtheory): P:=proc(q) local n;
for n from 1 to q do
if phi(sigma(n)-phi(n)-tau(n))=n then print(n);
fi; od; end: P(10^10);
  • Mathematica
    Select[Range[10^4], EulerPhi[DivisorSigma[1, #] - EulerPhi[#] - DivisorSigma[0, #]] == # &] (* Alonso del Arte, Jun 18 2014 *)
  • PARI
    isok(n) = (spd = sigma(n) - eulerphi(n) - numdiv(n)) && (eulerphi(spd) == n); \\ Michel Marcus, Jun 18 2014

Extensions

a(15)-a(16) from Jens Kruse Andersen, Jul 23 2014
a(17)-a(21) from Amiram Eldar, Jan 27 2019
Showing 1-7 of 7 results.

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