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.

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A373435 Iterate the function x <- phi(sigma(x)). The sequence lists the smaller member of cycles of length 2.

Original entry on oeis.org

4, 48, 72, 432, 1728, 10368, 184320, 1658880, 6220800, 10222080, 12856320000
Offset: 1

Views

Author

Jud McCranie, Jun 06 2024

Keywords

Comments

A cycle of length 2 also starts at 3852635996160. 3852635996160, 4869303828480, and 23971865863680 are also terms in the sequence. The sequence is complete through 10^13. - Jud McCranie, Sep 14 2024
166144927334400, 273145872384000, 1904394240000000,2779315686604800, 3644668394864640, 32729712349340160, 48693038284800000, 86790832128000000, 382404221337600000, 2684203735449600000, 5246585916751872000, 6169596402106368000, 13477567109529600000, 22998695842676736000, 38039819551128944640, 90555444080640000000, 102336861080974786560, 130026464870400000000, 222489728778240000000, 499064687988572160000, 2927044657152000000000, 19697331219625672704000, 23473340597403648000000, 73262977439150112768000, 1362680919097344000000000, 14128156119169341849600000, 16615689577928023080960000, 53129683677797469388800000, 6512790537509850316800000000, 125020570798295875584000000000, 201603700212193346715648000000, 1622429777898127409283072000000, 2631371767787268127693209600000, 71803515676046099742720000000000, 105852742809627160240717824000000000, 5528044915051901005564508897280000000, 15042880212263420006968149934080000000, 2013381648407800940932784726212608000000, 67868597277402193009117012867153920000000, 17285817653863442809402049534361600000000000 are also in this sequence. - Richard R. Forberg, Oct 27 2024

Examples

			phi(sigma(4)) = 6 and phi(sigma(6)) = 4, so 4 (the smallest term) is in the sequence.

Crossrefs

Subsequence of A067883 and A376256.
Cf. A000010, A000203, A062401, A001229, A095955, A095956, A373453, A373454.

Programs

  • Mathematica
    Select[Range[10^6], # == EulerPhi[DivisorSigma[1,EulerPhi[DivisorSigma[1,#]]]] && # < EulerPhi[DivisorSigma[1,#]]&] (* Stefano Spezia, Jun 07 2024 *)
  • PARI
    isok(x) = my(y = eulerphi(sigma(x))); if (y > x, x == eulerphi(sigma(y))); \\ Michel Marcus, Jun 06 2024

A373453 Iterate the function x <- phi(sigma(x)). The sequence has the smallest member of cycles of length 3.

Original entry on oeis.org

16, 1200, 15552, 67392, 272160, 69672960000
Offset: 1

Views

Author

Jud McCranie, Jun 06 2024

Keywords

Comments

69672960000 is also a term in the sequence.
a(7) <= 2704853606400. The numbers 242595672883200000, 66217181184000000000 and 185577469193591193600 are also terms. - Giorgos Kalogeropoulos, Jun 18 2024
4672651788288000 is also a term. - Jud McCranie, Jun 18 2024
The sequence is complete through 10^13. - Jud McCranie, Sep 14 2024

Examples

			16 -> 30 -> 24 -> 16, so 16 (the smallest term) is in the sequence.

Links

Crossrefs

Subsequence of A376256.
Cf. A001229, A095955, A095956, A373435, A373454.

Programs

  • PARI
    isok(x) = my(y = eulerphi(sigma(x))); if (y > x, my(z = eulerphi(sigma(y))); if (z > x, x == eulerphi(sigma(z)))); \\ Michel Marcus, Jun 07 2024

Extensions

a(6) from Giorgos Kalogeropoulos, Jun 18 2024

A373454 Iterate the function x <- phi(sigma(x)). The sequence has the smallest member of cycles of length 4.

Original entry on oeis.org

576, 41472, 2142720000, 3233260800
Offset: 1

Views

Author

Jud McCranie, Jun 06 2024

Keywords

Comments

130767436800000 is also a term. - Jud McCranie, Jun 18 2024
Terms are complete to 10^13. - Jud McCranie, Sep 14 2024
Terms also include 2590533833653034680320, 4911428805164059852800, 345401330417459527680000, 45369029282941832999731200, 1178793806496987670275686400000, 1241573383607207067648000000000, 3740981970485927435304960000000. - Richard R. Forberg, Oct 06 2024
Terms also include 1733855546435861719195867542454272000000. - Richard R. Forberg, Jan 04 2025

Examples

			576 -> 1512 -> 1280 -> 864 -> 576, so 576 (the smallest term) is in the sequence.

Crossrefs

Subsequence of A376256.
Cf. A000010, A000203, A001229, A095955, A095956, A373435, A373453.

Programs

  • PARI
    isok(x) = my(y = eulerphi(sigma(x))); if (y > x, my(z = eulerphi(sigma(y))); if (z>x, x == eulerphi(sigma(eulerphi(sigma(z)))))); \\ Michel Marcus, Jun 07 2024

A096850 Consider iteration of the function f(x) = phi(sigma(x)) = A062401(x). Sequence gives numbers n such that the trajectory of n returns to n.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 16, 24, 30, 48, 60, 72, 96, 128, 240, 432, 480, 576, 720, 864, 1200, 1280, 1512, 1536, 1728, 1800, 1860, 2016, 2560, 2880, 3024, 3456, 3840, 6912, 10368, 14080, 15552, 15840, 18144, 27648, 30976, 32768, 34560, 41472, 42240, 48384
Offset: 1

Views

Author

Labos Elemer, Jul 16 2004

Keywords

Examples

			Examples of cycles: {[1], [2], [4, 6], [8], [12], [16, 30, 24], [48, 60], [72, 96], [128]}.
95 => 32 => 36 => 72 => 96 => 72 => ..., therefore 72 and 96 are in the sequence.

Links

Crossrefs

Cf. A062401, A095952-A095956, A096887-A096890, A096849-A096851.

Programs

  • Mathematica
    a = {}; f[n_] := EulerPhi[ DivisorSigma[ 1, n]]; Do[ AppendTo[a, NestWhileList[f, n, UnsameQ, All][[ -1]]]; a = Union[a], {n, 10^6}]; Take[ a, 46] (* Robert G. Wilson v, Jul 21 2004 *)
  • PARI
    f(n)=eulerphi(sigma(n))
is(n)=my(t=f(n),h=f(t));while(t!=h,t=f(t);h=f(f(h));if(t==n, return(1)));t==n \\ Charles R Greathouse IV, Nov 27 2013

Extensions

Edited and extended by Robert G. Wilson v, Jul 21 2004

A096849 If f(x) = phi(sigma(x)) is iterated starting from these numbers, then the start-value never returns. These are the transient terms of this iteration. Never occur in terminal cycles.

Original entry on oeis.org

3, 5, 7, 9, 10, 11, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83
Offset: 1

Views

Author

Labos Elemer, Jul 16 2004

Keywords

Examples

			All odd and certain even integers belong here.

Links

Crossrefs

Cf. A095952, A095953, A095954, A095955, A095956, A096887, A096888, A096889, A096890, A096850.

Programs

  • Mathematica
    Flatten@ Table[Function[s, If[Length@ # > 0, First@ #, #] &@ Keys@ KeySelect[s, Length@ Lookup[s, #] == 1 &]]@ PositionIndex@ NestList[EulerPhi@ DivisorSigma[1, #] &, n, 10^2], {n, 71}] (* Michael De Vlieger, Jul 24 2017 *)

A375012 A cycle of length 5 when iterating x <- phi(sigma(x)).

Original entry on oeis.org

339026688000000, 377975808000000, 424639621324800, 483184764518400, 453984583680000
Offset: 1

Views

Author

Jud McCranie, Jul 27 2024

Keywords

Comments

The first term in the sequence is the smallest member of the cycle. This is the 5-cycle with the smallest members. Similar cycles are in the crossreferences.

Examples

			phi(sigma(339026688000000)) = 377975808000000, ... phi(sigma(453984583680000)) = 339026688000000, so 339026688000000 is in the sequence.

Crossrefs

Cf. A095955, A095956, A001229, A373435, A373453, A373454, A375013, A375014.

A375013 A cycle of length 6 when iterating x <- phi(sigma(x)).

Original entry on oeis.org

1800, 2880, 3024, 3840, 3456, 2560
Offset: 1

Views

Author

Jud McCranie, Jul 27 2024

Keywords

Comments

The first term in the sequence is the smallest member of the cycle. This is the 6-cycle with the smallest members. Similar cycles are in the crossreferences.

Examples

			phi(sigma(1800)) = 2880, ... phi(sigma(2560)) = 1800, so 1800 is in the sequence.

Crossrefs

Cf. A095955, A095956, A001229, A373435, A373453, A373454, A375012, A375014.

A375014 A cycle of length 6 when iterating x <- phi(sigma(x)).

Original entry on oeis.org

27648, 30976, 54432, 48384, 55296, 34560
Offset: 1

Views

Author

Jud McCranie, Jul 27 2024

Keywords

Comments

The first term in the sequence is the smallest member of the cycle. This is the 6-cycle with the second-smallest members. Similar cycles are in the crossreferences.

Examples

			phi(sigma(27648)) = 30976, ... phi(sigma(34560)) = 27648, so 27648 is in the sequence.

Crossrefs

Cf. A095955, A095956, A001229, A373435, A373453, A373454, A375012, A375013.

A096851 Even transient values of f(x)=phi(sigma(x)) iterations.

Original entry on oeis.org

10, 14, 18, 20, 22, 26, 28, 32, 34, 36, 38, 40, 42, 44, 46, 50, 52, 54, 56, 58, 62, 64, 66, 68, 70, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 98, 100, 102, 104, 106, 108, 110, 112, 114, 116, 118, 120, 122, 124, 126
Offset: 1

Views

Author

Labos Elemer, Jul 16 2004

Keywords

Links

Crossrefs

Even terms of A096849.
Cf. A095952-A095956, A096887-A096890, A096849-A096850.

A096996 a(n) is the smallest initial value if function f(x)=sigma(phi(x))=A062402(x) is iterated and the iteration ends in a cycle of length n.

Original entry on oeis.org

1, 5, 17, 401
Offset: 1

Views

Author

Labos Elemer, Jul 19 2004

Keywords

Comments

a(5) exists and <= 2^80.
a(6) = 883, a(11) = 88897, a(15) = 470137, a(18) = 448181;
a(9) = 24877841, a(12) = 16295171, a(21) <= 726569551. 254808457 does not reach a cycle after 14000 iterations. - Hiroaki Yamanouchi, Sep 06 2014
a(5) <= 9248288975491. - Hiroaki Yamanouchi, Sep 10 2014
a(5) > 8.5*10^7. - Tyler Busby, Mar 29 2024

Examples

			a(4) = 401. 401 -> 961 -> 2304 -> 2044 -> 2520 -> 1651 -> 4800 -> 3066 -> 2520 (cycle length = 4). - _Hiroaki Yamanouchi_, Sep 06 2014

Crossrefs

Cf. A062402, A095956.
Previous Showing 11-20 of 22 results. Next

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