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.

User: Jud McCranie

Jud McCranie's wiki page.

Jud McCranie has authored 398 sequences. Here are the ten most recent ones:

A383890 Index of record gaps between totient numbers.

Original entry on oeis.org

1, 2, 7, 30, 85, 257, 1031, 2493, 3288, 7604, 13392, 22663, 26818, 31377, 110175, 186971, 400432, 890621, 1536566, 17176199, 27501485, 102834105, 173246634, 182261294, 214104745, 268935021, 1781734397, 4010389565, 6213586719
Offset: 1

Views

Author

Jud McCranie, May 13 2025

Keywords

Comments

Values of k such that A002202(k+1) - A002202(k) > A002202(i+1) - A002202(i) for all iA383889.

Examples

			A002202(8)-A002202(7) = 4, which is greater than A002202(i+i)-A002202(i) for all i<7, so 7 is in the sequence.

Crossrefs

Cf. A000010, A002202, A083533, A383889.

A383889 Record high points in A083533.

Original entry on oeis.org

1, 2, 4, 6, 10, 12, 16, 18, 20, 22, 24, 26, 28, 32, 36, 40, 44, 50, 60, 64, 72, 74, 76, 78, 80, 90, 96, 108, 112
Offset: 1

Views

Author

Jud McCranie, May 13 2025

Keywords

Comments

Values of A083533(k) such that A083533(k) > A083533(i) for any iA002202(j+1) - A002202(j) > A002202(i+1) - A002202(i) for all iA383890 has the corresponding index of A083533.

Crossrefs

Cf. A000010, A002202, A083533, A383890.

A375122 A cycle of length 5 when iterating x <- phi(sigma(x)).

Original entry on oeis.org

6634509269055173050761216000, 7521613519844726223667200000, 7946886558074859593662464000, 7794495412499746337587200000, 7970172471593905204651622400
Offset: 1

Views

Author

Jud McCranie, Jul 30 2024

Keywords

Comments

The first term is the smallest member of the cycle. Another cycle with length 5 is A375012.

Crossrefs

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

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.

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.

A373739 a(n) is the smallest number that is in a cycle of length n when iterating x <- phi(sigma(x)).

Original entry on oeis.org

1, 4, 16, 576
Offset: 1

Views

Author

Jud McCranie, Jun 15 2024

Keywords

Comments

Terms in a cycle are distinct, i.e., going twice around a cycle of length 3 is not considered a cycle of length 6.
Other terms: a(5) <= 339026688000000, a(6) = 1800, a(8) <= 12959170560000, a(9) = 113218560, a(11) = 326592, a(12) = 5033164800, a(15) = 40255488, a(18) <= 150493593600, a(21) = 12227604480, a(22) <= 1316603904000.

Examples

			16 -> 24 -> 30 -> 16, 16 is the smallest member of any cycle of length 3, so a(3)=16.

Crossrefs

Cf. A062401, A095955, A095956, A373435, A373453, A373454.

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

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

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