A062402 -id:A062402 - cp's OEIS frontend

A096866 Function A062402(x) = sigma(phi(x)) is iterated. Starting with n, a(n) is the smallest term arising in trajectory, either in transient or in terminal cycle.

1, 1, 3, 3, 5, 3, 7, 7, 7, 7, 7, 7, 13, 7, 15, 15, 17, 7, 19, 15, 21, 7, 23, 15, 25, 26, 27, 28, 29, 15, 31, 31, 28, 31, 31, 28, 37, 31, 31, 31, 31, 28, 43, 28, 31, 28, 31, 31, 49, 28, 51, 31, 53, 31, 31, 31, 57, 31, 31, 31, 61, 31, 63, 63, 65, 28, 67, 63, 31, 31, 71, 31, 73, 74

Offset: 1

Labos Elemer, Jul 21 2004

nonn

			n=240: list={240,127,312,[252,195],252,...}, a(240)=127, a transient;
n=254: list={254,312,[252,195],252,...}, a(254)=195, a recurrent term.

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

Cf. A062401, A062402, A096862, A096863, A096864 (largest term), A096993.

Cf. also A096865.

gf[x_] :=DivisorSigma[1, EulerPhi[x]] gite[x_, hos_] :=NestList[gf, x, hos] Table[Min[gite[w, 20]], {w, 1, 256}]

(define (A096866 n) (let loop ((visited (list n)) (m n)) (let ((next (A062402 (car visited)))) (cond ((member next visited) m) (else (loop (cons next visited) (min m next))))))) ;; Antti Karttunen, Nov 18 2017

A096993 Function A062402(x) = sigma(phi(x)) is iterated with initial value=n. a(n) is the length of cycle into which the trajectory merges.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 2, 2, 1, 2, 1, 1, 3, 2, 3, 1, 1, 2, 1, 1, 1, 1, 3, 1, 3, 1, 3, 3, 1, 3, 3, 1, 2, 3, 3, 3, 3, 1, 2, 1, 3, 1, 3, 3, 2, 1, 2, 3, 2, 3, 3, 3, 2, 3, 3, 3, 2, 3, 2, 2, 2, 1, 2, 2, 3, 3, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 2, 3, 2, 3, 2, 2, 2, 3, 2, 3, 2, 3, 2, 3, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2

Offset: 1

Labos Elemer, Jul 19 2004

nonn

No 5's present among the first 16384 terms, but they should exist as A095955 has them too. - Antti Karttunen, Dec 04 2017

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

Cf. A062402, A095955, A096864, A096866.

(define (A096993 n) (if (= 1 n) n (let loop ((visited (list n)) (i 1)) (let ((next (A062402 (car visited)))) (cond ((member next visited) => (lambda (prepath) (+ 1 (- i (length prepath))))) (else (loop (cons next visited) (+ 1 i)))))))) ;; Antti Karttunen, Dec 04 2017

A096864 Function A062402(x) = sigma(phi(x)) is iterated. Starting with n, a(n) is the largest term arising in trajectory, either in transient or in terminal cycle.

Original entry on oeis.org

1, 2, 3, 4, 12, 6, 12, 12, 12, 12, 18, 12, 28, 14, 15, 16, 72, 18, 72, 20, 28, 22, 36, 24, 42, 28, 72, 28, 72, 30, 72, 72, 42, 72, 72, 36, 252, 72, 72, 72, 90, 42, 252, 44, 72, 46, 72, 72, 252, 50, 252, 72, 252, 72, 90, 72, 252, 72, 90, 72, 168, 72, 252, 252, 168, 66, 168, 252

Offset: 1

Labos Elemer, Jul 21 2004

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

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

Cf. A062401, A062402, A066437, A096862, A096863, A096866 (smallest term), A096993.

Cf. also A096861.

gf[x_] :=DivisorSigma[1, EulerPhi[x]] gite[x_, hos_] :=NestList[gf, x, hos] Table[Max[gite[w, 20]], {w, 1, 256}]
Table[Max[NestList[DivisorSigma[1,EulerPhi[#]]&,n,20]],{n,70}] (* Harvey P. Dale, May 13 2019 *)

(define (A096864 n) (let loop ((visited (list n)) (m n)) (let ((next (A062402 (car visited)))) (cond ((member next visited) m) (else (loop (cons next visited) (max m next))))))) ;; Antti Karttunen, Nov 18 2017

a(n) = max(n, A066437(n)). - Antti Karttunen, Dec 06 2017

A096854 a(n) = A062402(2^n-1).

Original entry on oeis.org

1, 3, 12, 15, 72, 91, 312, 255, 1240, 1860, 4123, 5080, 26208, 34200, 93600, 65535, 334368, 416560, 1420800, 1596364, 6146800, 5949696, 20485332, 23788842, 120519630, 194016600, 358132380, 458803800, 1674738000, 2166798816, 6045990912, 4294967295, 22739738112, 37862623140

Offset: 1

Labos Elemer, Jul 19 2004

nonn

Amiram Eldar, Table of n, a(n) for n = 1..400

Cf. A000203, A053287, A062401, A062402, A053287, A096853, A096854, A096855, A096856.

Table[DivisorSigma[1, EulerPhi[2^n - 1]], {n, 1, 30}]

a(n) = sigma(eulerphi(2^n-1)); \\ Michel Marcus, Aug 30 2019

a(n) = A000203(A053287(n)). - Amiram Eldar, Jun 04 2024

More terms from Michel Marcus, Aug 30 2019

A096856 a(n) = A062402(2^n+1).

Original entry on oeis.org

1, 3, 7, 12, 31, 42, 124, 224, 511, 847, 1953, 2688, 12264, 18816, 29127, 72540, 131071, 195048, 558523, 1077440, 3164112, 4552020, 10890040, 10342080, 54525848, 73260781, 155671040, 318848400, 1080311232, 964580240, 3070642080, 4340711424, 13722819600, 19039027200

Offset: 0

Labos Elemer, Jul 19 2004

nonn

Amiram Eldar, Table of n, a(n) for n = 0..372

Cf. A000203, A053285, A062401, A062402, A053287, A096853, A096854, A096855.

Table[DivisorSigma[1, EulerPhi[2^n + 1]], {n, 0, 30}]

a(n) = A000203(A053285(n)). - Amiram Eldar, Jun 04 2024

Offset changed to 0, a(0) prepended and three more terms added by Amiram Eldar, Jun 04 2024

A096862 Function A062402(x)=sigma(phi(x)) is iterated. Starting with n, a(n) is the count of distinct terms arising during this trajectory; a(n)=t(n)+c(n)=t+c, where t is the number of transient terms, c is the number of recurrent terms [in the terminal cycle].

Original entry on oeis.org

1, 2, 1, 2, 3, 2, 2, 3, 3, 3, 4, 2, 2, 3, 1, 2, 4, 3, 5, 2, 2, 4, 3, 2, 3, 2, 5, 1, 5, 2, 3, 4, 3, 4, 4, 2, 4, 5, 4, 4, 5, 2, 6, 3, 4, 3, 4, 4, 6, 3, 5, 4, 7, 5, 5, 4, 4, 5, 5, 3, 3, 4, 4, 5, 3, 3, 4, 5, 5, 4, 4, 3, 3, 4, 5, 4, 3, 4, 3, 5, 6, 5, 5, 4, 5, 6, 6, 5, 4, 4, 3, 5, 3, 4, 3, 5, 3, 6, 3, 5, 8, 5, 4, 3, 3

Offset: 1

Labos Elemer, Jul 21 2004

nonn

			n=256: list={256,255,255}, transient=t=1, cycle=c=1, a(256)=t+c=2.

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

gf[x_] :=DivisorSigma[1, EulerPhi[x]] gite[x_, hos_] :=NestList[gf, x, hos] Table[Length[Union[gite[w, 1000]]], {w, 1, 256}]

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

Original entry on oeis.org

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).

A096995 Number of transient terms if f(x) = sigma(phi(x)) = A062402 is iterated at initial value = 2^n.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 3, 3, 1, 2, 3, 5, 2, 3, 6, 15, 1, 6, 8, 3, 15, 9, 4, 65, 44, 82, 83, 77, 75, 48, 26, 43, 1

Offset: 0

Labos Elemer, Jul 22 2004

For transient lengths of iterations A062401(x) or A062402(x), if started at 2^n, holds that A096994(n)+1 = a(n). Corresponding cycle lengths satisfy A096852(n-1) = A096857(n). Behind these observation several relationships stand, e.g., sigma(A062401(x)) = A062402(sigma(x)) or phi(A062402(x)) = A062401(phi(x)).

For initial value = 2^33 more than 38000 iterations did not lead to a recurrent term, so possibly there is no cycle. a(34) through a(39) are 8, 52, 71, 24, 40, 12. - Klaus Brockhaus, Jul 19 2007

			Trajectory of 2^0 is 1,1, ...; there are zero transient terms preceding the 1-cycle (1), so a(0) = 0.
Trajectory of 2^14 is 16384, 16383, 34200, 30480, 26520, 16380, 10200, 6138, 6045, 9906, 9920, 12264, 10200, ...; there are six transient terms preceding the 6-cycle (10200, 6138, 6045, 9906, 9920, 12264), so a(14) = 6.

Cf. A062402, A062401, A096857, A096852, A096994.

With[{nn = 10^4}, Table[Count[Values@ PositionIndex@ NestList[DivisorSigma[1, EulerPhi@ #] &, 2^n, nn], ?(Length@ # == 1 &)], {n, 0, 60}] /. m /; m == nn + 1 -> -1] (* Michael De Vlieger, Jul 24 2017 *)

Edited and corrected by Klaus Brockhaus, Jul 19 2007

A065390 Peak values reached by A062402 at the sites listed in A065389.

Original entry on oeis.org

1, 3, 7, 12, 18, 28, 31, 39, 42, 56, 72, 91, 96, 98, 168, 195, 252, 280, 312, 360, 372, 392, 546, 576, 744, 840, 864, 992, 1092, 1170, 1344, 1512, 1680, 1860, 1872, 2016, 2240, 2418, 2880, 3224, 3600, 3844, 4320, 4368, 4914, 5082, 5952, 6045, 6552, 7440

Offset: 1

Labos Elemer, Nov 05 2001

nonn

Amiram Eldar, Table of n, a(n) for n = 1..958 (terms 1..500 from Harry J. Smith)

Cf. A062402, A000203, A000010, A065389, A065391, A065392, A065393, A065394.

a=0; s=0; Do[s=DivisorSigma[1, EulerPhi[n]]; If[s>a, a=s; Print[s]], {n, 1, 10000}]
(* Second program: *)
Union@ FoldList[Max, Array[DivisorSigma[1, EulerPhi[#]] &, 2200]] (* Michael De Vlieger, Jun 19 2018 *)

{ n=r=0; for (m=1, 10^9, x=sigma(eulerphi(m)); if (x > r, r=x; write("b065390.txt", n++, " ", x); if (n==500, return)) ) } \\ Harry J. Smith, Oct 18 2009

a(n) = A062402(A065389(n)). - Amiram Eldar, Mar 22 2025

A096988 Initial values for f(x)=sigma(phi(x))=A062402(x) such that iteration of f ends in cycle of length=1.

Original entry on oeis.org

1, 2, 3, 4, 6, 13, 15, 16, 20, 21, 23, 24, 25, 26, 28, 30, 33, 36, 42, 44, 46, 50, 66, 157, 169, 203, 215, 237, 241, 245, 255, 256, 261, 272, 275, 287, 303, 305, 314, 316, 320, 325, 338, 340, 344, 347, 367, 369, 375, 384, 385, 392, 393, 404, 406, 408, 429, 430

Offset: 1

Labos Elemer, Jul 19 2004

nonn

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A062402, A095952, A018784.

a = {}; Do[s = {n}; While[! MemberQ[s, k = DivisorSigma[1, EulerPhi[s[[-1]]]]], AppendTo[s, k]]; If[s[[-1]] == k, AppendTo[a, n]], {n, 430}]; a (* Ivan Neretin, Dec 15 2016 *)

cp's OEIS Frontend

A096866 Function A062402(x) = sigma(phi(x)) is iterated. Starting with n, a(n) is the smallest term arising in trajectory, either in transient or in terminal cycle.

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A096993 Function A062402(x) = sigma(phi(x)) is iterated with initial value=n. a(n) is the length of cycle into which the trajectory merges.

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A096864 Function A062402(x) = sigma(phi(x)) is iterated. Starting with n, a(n) is the largest term arising in trajectory, either in transient or in terminal cycle.

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A096854 a(n) = A062402(2^n-1).

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A096856 a(n) = A062402(2^n+1).

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A096862 Function A062402(x)=sigma(phi(x)) is iterated. Starting with n, a(n) is the count of distinct terms arising during this trajectory; a(n)=t(n)+c(n)=t+c, where t is the number of transient terms, c is the number of recurrent terms [in the terminal cycle].

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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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A096995 Number of transient terms if f(x) = sigma(phi(x)) = A062402 is iterated at initial value = 2^n.

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A065390 Peak values reached by A062402 at the sites listed in A065389.

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