A095955 -id:A095955 - cp's OEIS frontend

A096863 Function A062402(x)=sigma(phi(x)) is iterated. Starting with n, a(n) is the count of transient terms of trajectory.

0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 2, 0, 1, 1, 0, 1, 1, 1, 2, 1, 1, 2, 2, 1, 2, 1, 2, 0, 2, 1, 0, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 4, 2, 1, 2, 1, 1, 4, 2, 3, 1, 5, 2, 2, 1, 2, 2, 2, 0, 1, 1, 2, 3, 1, 2, 2, 3, 2, 1, 2, 0, 1, 2, 2, 2, 1, 1, 1, 3, 4, 2, 3, 1, 3, 4, 4, 2, 2, 1, 1, 2, 1, 1, 1, 3, 1, 4, 1, 2, 6, 3, 2, 1, 1

Offset: 1

Labos Elemer, Jul 21 2004

nonn

a(n)=0 means that n is a recurrent term from A096998.

			n=256: list={256,255,255}, a(256)=1;
n=101: list={101,217,546,403,1170,819,[1240,1512],1240,...,a(101)=6;

Cf. A062401, A062402, A095955, A096859-A096866, A096993.

a(n) = A096861(n)-A096993(n).

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

A036840 a(n) is the average of the repeating terms of {T(n,k)} rounded to the nearest integer (rounding up if there's a choice), if {T(n,k)} is eventually periodic; = 0 otherwise. Here T(n,k) is the "phi/sigma tug-of-war sequence with seed n" defined by T(n,1) = phi(n), T(n,2) = sigma(phi(n)), T(n,3) = phi(sigma(phi(n))), ..., T(n,k) = phi(T(n,k-1)) if k is odd and = sigma(T(n,k-1)) if k is even.

Original entry on oeis.org

1, 1, 3, 3, 7, 3, 7, 7, 7, 7, 7, 7, 20, 7, 12, 12, 39, 7, 39, 12, 20, 7, 20, 12, 20, 20, 39, 20, 39, 12, 39, 39, 20, 39, 39, 20, 154, 39, 39, 39, 39, 20, 154, 20, 39, 20, 39, 39, 154, 20, 154, 39, 154, 39, 39, 39, 154, 39, 39, 39, 100, 39, 154, 154, 100, 20, 100, 154, 39, 39

Offset: 1

Joseph L. Pe, Jan 09 2002

nonn

Conjecture: a(n) is never 0; i.e. the sequence {T(n,k)} is eventually periodic for every n.

a(n) - n can be thought of as the final score in the phi/sigma tug-of-war with seed n. For example a(5) - 5 = 7 - 5 = 2, so sigma wins by "2 points" over phi at 5. a(8) - 8 = 7 - 8 = -1, so phi wins by "1 point" over sigma at 8. a(3) - 3 = 3 - 3 = 0, so it is a tie at 3. Are sigma's margins of victory over phi bounded? Are phi's bounded?

			The sequence {T(5,k)} is 4, 7, 6, 12, 4, 7, .... The average of the repeating numbers is 7.25 which rounds off to 7. So a(5) = 7. The sequence {T(37,k)} is 36, 91, 72, 195, 96, 252, 72, 195, .... The average of the repeating numbers is 153.75, which rounds off to 154. So a(37) = 154.

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A000010, A000203, A036845, A066437, A095955, A096993.

a[ n_ ] := Module[ {}, For[ m=n; seq={}, !MemberQ[ seq, m ], m=DivisorSigma[ 1, EulerPhi[ m ] ], AppendTo[ seq, m ] ]; rp=Drop[ seq, Position[ seq, m ][ [ 1, 1 ] ]-1 ]; Floor[ 1/2+(Plus@@Join[ rp, EulerPhi/@rp ])/2/Length[ rp ] ] ]

(define (A036840 n) (let loop ((visited (list n)) (i 1)) (let ((next ((if (odd? i) A000010 A000203) (car visited)))) (cond ((member next (reverse visited)) => (lambda (start_of_cyclic_part) (cond ((even? (length start_of_cyclic_part)) (floor->exact (+ 1/2 (/ (apply + start_of_cyclic_part) (length start_of_cyclic_part))))) (else (loop (cons next visited) (+ 1 i)))))) (else (loop (cons next visited) (+ 1 i))))))) ;; Antti Karttunen, Dec 06 2017

Edited by Dean Hickerson, Jan 18 2002

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.

A097007 a(n) = index of first appearance of n in A096859.

Original entry on oeis.org

1, 3, 7, 18, 34, 52, 100, 422, 882, 1008, 960, 912, 784, 1497, 3187, 13456, 21336, 42682, 69696, 50176, 73191, 112896, 88452, 151828, 140736, 198876, 245028, 187272, 252964, 207936, 229456, 447201, 1412589, 9734400, 7757136, 7910076

Offset: 1

Labos Elemer, Jul 26 2004

nonn

a(n) = smallest k such that A096860(k) + A095955(k) = n.

a(n) = smallest k such that n equals the index of the term that completes the first cycle in the trajectory of k under iteration of f(x) = A062401(x) = phi(sigma(x)).

			The trajectory of 18 under iteration of f(x) is 18, 24, 16, 30, 24, 16, 30, ...; the cycle (24, 16, 30) is completed at the fourth term and for j < 18 the first cycle in trajectory of j under iteration of f(x) is completed at the first, second or third term, hence a(4) = 18.
The trajectory of 69696 under iteration of f(x) is 69696, 163296, 157248, 193536, 247808, 217728, 147456, 324000, 285120, 332640, 331776, 900900, 967680, 991232, 1143072, 2122848, 2201472, 1658880, 1801800, 1658880, 1801800, ...; the cycle (1658880, 1801800) is completed at the 19th term and for j < 69696 the first cycle in trajectory
of j under iteration of f(x) is completed at an earlier term, hence a(19) = 69696.

Klaus Brockhaus, Table of n, a(n) for n=1..119

Cf. A062401, A096859, A096860, A095955, A097008.

fs[x_] :=EulerPhi[DivisorSigma[1, x]]; nsf[x_, ho_] :=NestList[fs, x, ho]; luf[x_, ho_] :=Length[Union[nsf[x, ho]]] t=Table[0, {35}]; Do[s=luf[n, 100]; If[s<36&&t[[s]]==0, t[[s]]=n], {n, 1, 1600000}]; t

{v=vector(40); for(n=1, 10000000, k=n; s=Set(k); until(setsearch(s, k=eulerphi(sigma(k))), s=setunion(s, Set(k))); a=#s; if(a<=m&&v[a]==0, v[a]=n)); v} /* Klaus Brockhaus, Jul 16 2007 */

Edited, a(27) corrected and a(34) through a(36) added by Klaus Brockhaus, Jul 16 2007

A032450 Period of finite sequence g(n) related to Poulet's Conjecture.

Original entry on oeis.org

1, 3, 2, 2, 3, 7, 6, 12, 4, 2, 3, 12, 4, 7, 6, 4, 7, 6, 12, 15, 8, 12, 28, 6, 12, 4, 7, 12, 4, 7, 6, 28, 12, 6, 12, 4, 7, 8, 15, 8, 15, 31, 30, 72, 24, 60, 16, 6, 12, 4, 7, 24, 60, 16, 31, 30, 72, 8, 15, 12, 28, 16, 31, 30, 72, 24, 60, 12, 28, 8, 15, 60, 16

Offset: 1

Ursula Gagelmann (gagelmann(AT)altavista.net), Apr 07 1998

nonn

Poulet's Conjecture states that for any integer n, the sequence f_0(n) = n, f_2k+1(n)=sigma(f_2k(n)), f_2k(n)=phi(f_2k-1(n)) (where sigma = A000203 and phi = A000010) is eventually periodic.

			Poulet's sequence starting at 1 is 1->1->1->.. which contributes [1]. Starting at 2: 2->3->2->3->.. which contributes [3,2]. Starting at 3: 3->4->2->3->2->3... which contributes [2,3]. Starting at 4: 4->7->6->12->4->7->6->12.. which contributes  [7, 6, 12, 4]. - _R. J. Mathar_, May 08 2020

P. Poulet, Nouvelles suites arithmétiques, Sphinx vol. 2 (1932) pp. 53-54.

Leon Alaoglu and Paul Erdős, A conjecture in elementary number theory, Bull. Amer. Math. Soc. 50 (1944), 881-882.
Sean A. Irvine, Java program (github)

Cf. A000010, A000203, A001229, A036845, A095955, A096865.

g(1)=n; thereafter g(2k)=sigma(g(2k-1)), g(2k+1)=phi(g(2k)).

Revised definition and added formula from Ursula Gagelmann, Apr 07 1998 - N. J. A. Sloane, May 08 2020

Missing a(42)=31 inserted and more terms from Sean A. Irvine, Jun 21 2020

A082991 a(1) = 1 and for n > 1, a(n) = 2 * length of the cycle reached for the map x -> A062401(x), starting at n [where A062401(n) = phi(sigma(n))], or -1 if no finite cycle is ever reached.

Original entry on oeis.org

1, 2, 2, 4, 2, 4, 4, 2, 2, 4, 4, 2, 4, 2, 2, 6, 4, 6, 2, 2, 6, 2, 2, 6, 6, 2, 6, 6, 2, 6, 6, 4, 6, 6, 6, 4, 6, 6, 6, 6, 2, 4, 2, 6, 6, 6, 6, 4, 4, 4, 6, 4, 6, 4, 6, 4, 4, 6, 6, 4, 6, 4, 4, 4, 6, 4, 4, 4, 4, 4, 6, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 6, 4, 4, 6, 4, 4, 6, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4

Offset: 1

Benoit Cloitre, May 29 2003

nonn

From the original definition: Define a sequence u_n as follows: u_n(1) = n, thereafter u_n(2k) = sigma(u_n(2k-1)), u_n(2k+1) = phi(u_n(2k)); then a(n) is the length of the ultimate period of u_n(k) (which is conjectured to become ultimately periodic for any n>=1).
Conjecture: despite results for small terms, all even number are obtained as values. (For example, 12 occurs since a(12102) = 12).
From Antti Karttunen, Nov 07 2017: (Start)
Because for all n > 1, A000010(n) < n and A062401(n) > 1, such sequences u_n cannot end in odd cycle when n > 1. From this follows that for n > 1, a(n) = 2 * length of the cycle reached for the map x->A062401(x), starting at n, or -1 if no finite cycle is ever reached.
See entry A095955 for further notes about the occurrence of cycles.
(End)

			If n=6, u(1)=6, u(2)=sigma(6)=12, u(3)=phi(12)=4, u(4)=sigma(4)=7 u(5)=phi(7)=6, hence u(k) becomes periodic with period (6,12,4,7) of length 4 and a(6)=4.

J. Berstel et al., Combinatorics on Words: Christoffel Words and Repetitions in Words, Amer. Math. Soc., 2008. See p. 83.

Antti Karttunen, Table of n, a(n) for n = 1..16385

Cf. A000010, A000203, A062401, A095955.

a(1) = 1; for n > 1, a(n) = 2*A095955(n). [See comments.] - Antti Karttunen, Nov 07 2017

Definition simplified by Antti Karttunen, Nov 07 2017

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.

cp's OEIS Frontend

A096863 Function A062402(x)=sigma(phi(x)) is iterated. Starting with n, a(n) is the count of transient terms of trajectory.

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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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A097007 a(n) = index of first appearance of n in A096859.

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A032450 Period of finite sequence g(n) related to Poulet's Conjecture.

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A082991 a(1) = 1 and for n > 1, a(n) = 2 * length of the cycle reached for the map x -> A062401(x), starting at n [where A062401(n) = phi(sigma(n))], or -1 if no finite cycle is ever reached.

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