A373435 -id:A373435 - cp's OEIS frontend

A001229 Numbers k such that phi(sigma(k)) = k.

1, 2, 8, 12, 128, 240, 720, 6912, 32768, 142560, 712800, 1140480, 1190400, 3345408, 3571200, 5702400, 14859936, 29719872, 50319360, 118879488, 2147483648, 3889036800, 4389396480, 21946982400, 47416320000, 92177326080, 133145026560, 331914240000, 460886630400, 665725132800

Offset: 1

N. J. A. Sloane

nonn

For n=0,1,2,3,4 & 5 2^(2^n-1) is in the sequence because 2^2^n+1 is prime for n=0,1,2,3 & 4 (Fermat primes). - Farideh Firoozbakht, Oct 08 2004

2987228160000, 11681629470720, and 47996928000000 are also in the sequence. - Jud McCranie, Sep 14 2024

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 128, p. 44, Ellipses, Paris 2008.
J.-M. De Koninck & A. Mercier, 1001 Problemes en Theorie Classique Des Nombres, Problem 702 pp. 92; 300-1, Ellipses Paris 2004.
R. K. Guy, Unsolved Problems in Number Theory, B42.

Leon Alaoglu and Paul Erdős, A conjecture in elementary number theory, Bull. Amer. Math. Soc. 50 (1944), pp. 881-882.
Graeme L. Cohen, On a conjecture of Makowski and Schinzel, Colloquium Mathematicae, Vol. 74, No. 1 (1997), pp. 1-8. See Notes p. 7.
Fred W. Helenius, 664 solutions [Broken Link]
Fred W. Helenius, 664 solutions [From the Wayback machine]
T. Negadi, The genetic code invariance: when Euler and Fibonacci meet, arXiv preprint arXiv:1406.6092 [q-bio.OT], 2014; Symmetry: Culture and Science, Vol. 25, No. 3, 261-278, 2014.
Eric Weisstein's World of Mathematics, Totient Function.

Cf. A000010, A018784, A135240 A373435, A373453, A373454.

Select[Range[10000], EulerPhi[DivisorSigma[1, #]] == # &] (* T. D. Noe, Jun 26 2012 *)

is(n)=eulerphi(sigma(n))==n \\ Charles R Greathouse IV, May 15 2013

phi(A018784), sorted. - David W. Wilson, Oct 18 2012

More terms from David W. Wilson, Aug 15 1996 (search was complete only through a(19) = 50319360).
Jud McCranie reports Jun 15 1998 that the terms through a(24) are certain.
a(28) added. Verified sequence is complete through a(28) by Donovan Johnson, Jun 30 2012
More terms from Jud McCranie, Sep 14 2024, Sep 14 2024. Complete through 10^13.

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

Jud McCranie, Jun 06 2024

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

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

Richard R. Forberg, Additional terms, Nov 2024.

Subsequence of A376256.

Cf. A001229, A095955, A095956, A373435, A373454.

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

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

Jud McCranie, Jun 06 2024

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

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

Subsequence of A376256.

Cf. A000010, A000203, A001229, A095955, A095956, A373435, A373453.

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

A376256 Numbers which are the minimum of a cycle in the map x -> phi(sigma(x)).

Original entry on oeis.org

1, 2, 4, 8, 12, 16, 48, 72, 128, 240, 432, 576, 720, 1200, 1728, 1800, 6912, 10368, 15552, 27648, 32768, 41472, 67392, 142560, 184320, 272160, 326592, 712800, 1140480, 1190400, 1658880, 3345408, 3571200, 5702400, 6220800, 10222080, 14859936, 29719872, 40255488, 50319360, 113218560, 118879488

Offset: 1

Richard R. Forberg, Sep 16 2024

nonn

No further terms < 254731536.
The status of 254731536 is unknown, but conjectured not a term.
Additional terms include 2142720000, 5033164800, 150493593600, 3852635996160.
See further cycles in the linked document here below which contains 422 cycles. It includes the 80 cycles complied by Jud McCrainie in a linked document at A095955.

Richard R. Forberg, The Prevalence of Cycles vs. "Unruly Events" from 10^8 to 10^55 after Iterating on n <- phi(sigma(n))
Richard R. Forberg, 422 Phi Sigma Cycles Organized by Length

Union of A001229, A373435, A373453, A373454, etc.

Cf. A062401, A095955.

\\ Naive program, assumes eventual termination (ok upto 254731535).
isok(n)={my(M=Map(),p=n); while(!mapisdefined(M,p) && p>=n, mapput(M,p,1); p=eulerphi(sigma(p))); p==n} \\ Andrew Howroyd, Sep 19 2024

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

Original entry on oeis.org

339026688000000, 377975808000000, 424639621324800, 483184764518400, 453984583680000

Offset: 1

Jud McCranie, Jul 27 2024

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.

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

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

Jud McCranie, Jul 27 2024

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.

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

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

Jud McCranie, Jul 27 2024

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.

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

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

A067883 Numbers k such that k = phi(sigma(phi(sigma(k)))).

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 48, 60, 72, 96, 128, 240, 432, 480, 720, 1728, 2016, 6912, 10368, 14080, 32768, 142560, 184320, 196560, 712800, 1140480, 1190400, 1658880, 1801800, 3345408, 3571200, 5702400, 6220800, 8363520, 10222080, 10368000, 14859936, 29719872, 50319360

Offset: 1

Benoit Cloitre, Mar 02 2002

This sequence has both terms of cycles of length 2 when iterating k <-phi(sigma(k)) (see A373435) plus cycles of length 1 (see A001229). - Jud McCranie, Aug 09 2024

Cf. A000010, A000203, A062401, A163372.

Supersequence of A001229. A373435 is a subsequence of this sequence.

Select[Range[10^6], EulerPhi[DivisorSigma[1, EulerPhi[DivisorSigma[1, #]]]] == # &] (* Amiram Eldar, Apr 24 2022 *)

More terms from Reiner Martin, Jul 09 2002

a(35)-a(39) from Amiram Eldar, Apr 24 2022

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

Jud McCranie, Jun 15 2024

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.

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

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

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

Original entry on oeis.org

6634509269055173050761216000, 7521613519844726223667200000, 7946886558074859593662464000, 7794495412499746337587200000, 7970172471593905204651622400

Offset: 1

Jud McCranie, Jul 30 2024

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

Cf. A095955, A095956, A001229, A373435, A373453, A373454, A375012, A375013, A375014.

cp's OEIS Frontend

A001229 Numbers k such that phi(sigma(k)) = k.

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

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

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A376256 Numbers which are the minimum of a cycle in the map x -> phi(sigma(x)).

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

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

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

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A067883 Numbers k such that k = phi(sigma(phi(sigma(k)))).

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

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