A095955 -id:A095955 - cp's OEIS frontend

A096852 a(n) is the length of terminal cycle of the trajectory of f(x)=phi(sigma(x)) if started at 2^n.

1, 1, 2, 1, 3, 2, 2, 1, 2, 2, 6, 2, 1, 6, 2, 1, 2, 3, 11, 11, 2, 2, 15, 15, 18, 18, 18, 18, 12, 12, 12, 1

Offset: 0

Labos Elemer, Jul 16 2004

nonn

			n=18: start = 262144 and the corresponding 11-cycle is 262144, 524286, [368640, 381024, 326592, 550368, 435456, 580608, 851840, 552960, 524160, 442368, 432000], 368640, ...

Cf. A095955, A095956, A096849, A096850.

g[n_] := EulerPhi[ DivisorSigma[1, n]]; f[n_] := Block[{lst = NestWhileList[g, n, UnsameQ, All]}, -Subtract @@ Flatten[ Position[lst, lst[[ -1]]]]]; Table[ f[2^n], {n, 0, 20}]

f(x)=eulerphi(sigma(x))
a(n)=my(t=f(2^n), h=f(t), s); while(t!=h, t=f(t); h=f(f(h))); t=f(t); h=f(t); s=1; while(t!=h, s++; t=f(t); h=f(f(h))); s \\ Charles R Greathouse IV, Nov 27 2013

a(n) = A095955(2^n). - Charles R Greathouse IV, Nov 27 2013

Edited, corrected and extended by Robert G. Wilson v, Jul 17 2004

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

Jud McCranie, Jun 06 2024

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

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

Subsequence of A067883 and A376256.

Cf. A000010, A000203, A062401, A001229, A095955, A095956, A373453, A373454.

Select[Range[10^6], # == EulerPhi[DivisorSigma[1,EulerPhi[DivisorSigma[1,#]]]] && # < EulerPhi[DivisorSigma[1,#]]&] (* Stefano Spezia, Jun 07 2024 *)

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

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

A096859 Function A062401(x) = phi(sigma(x)) = f(x) is iterated. Starting with n, a(n) is the count of distinct terms arising in trajectory; a(n)=t(n)+c(n)=t+c, where t=number of transient terms, c=number of recurrent terms (in the terminal cycle).

Original entry on oeis.org

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

Offset: 1

Labos Elemer, Jul 21 2004

nonn

			n=255: list={255,144,360,288,[432,480],432,...}, t=transient=4, c=cycle=2, a(255)=t+c=6;
n=244: list={244,180,144,360,288,[432,480],432,...}, t=5, c=2, a(244)=7.

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

Cf. A000010, A000203, A062401, A062402, A095955, A096860, A096861, A096865.

Cf. also A096862.

fs[x_] :=EulerPhi[DivisorSigma[1, x]] itef[x_, len_] :=NestList[fs, x, len] Table[Length[Union[itef[2^w, 20]]], {w, 1, 256}] (* len=20 at n<=256 is suitable *)

(define (A096859 n) (let loop ((visited (list n)) (i 1)) (let ((next (A062401 (car visited)))) (cond ((member next visited) i) (else (loop (cons next visited) (+ 1 i))))))) ;; Antti Karttunen, Nov 18 2017

A096865 Function A062401(x) = phi(sigma(x)) is iterated. Starting with n, a(n) is the smallest term arising in trajectory, either in transient or in terminal cycle.

Original entry on oeis.org

1, 2, 2, 4, 2, 4, 4, 8, 9, 4, 4, 12, 4, 8, 8, 16, 4, 16, 8, 12, 16, 12, 8, 16, 16, 12, 16, 16, 8, 16, 16, 32, 16, 16, 16, 36, 16, 16, 16, 16, 12, 32, 12, 16, 16, 16, 16, 48, 36, 48, 16, 32, 16, 32, 16, 32, 32, 16, 16, 48, 16, 32, 48, 64, 16, 48, 32, 36, 32, 48, 16, 72, 36, 36, 48

Offset: 1

Labos Elemer, Jul 21 2004

nonn

			n=255: list={255,144,360,288,[432,480],432,...}, a(255)=144 as a transient term;
n=254: list={254,[128],128,...}, a(254)=128, as a fixed point.

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

Cf. A062401, A062402, A095955, A096859, A096860, A096861 (largest term).

Cf. also A096866.

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

(define (A096865 n) (let loop ((visited (list n)) (m n)) (let ((next (A062401 (car visited)))) (cond ((member next visited) m) (else (loop (cons next visited) (min m next))))))) ;; Antti Karttunen, Nov 18 2017

A096861 Function A062401(x) = phi(sigma(x)) = f(x) is iterated. Starting with n, a(n) is the largest term arising in trajectory.

Original entry on oeis.org

1, 2, 3, 6, 5, 6, 7, 8, 12, 10, 11, 12, 13, 14, 15, 30, 17, 30, 19, 20, 30, 22, 23, 30, 30, 26, 30, 30, 29, 30, 31, 96, 33, 34, 35, 96, 37, 38, 39, 40, 41, 96, 43, 44, 45, 46, 47, 60, 96, 60, 51, 96, 53, 96, 55, 96, 96, 58, 59, 60, 61, 96, 63, 126, 65, 66, 96, 96, 96, 70, 71, 96

Offset: 1

Labos Elemer, Jul 21 2004

nonn

			n=255: list={255,144,360,288,[432,480],432,...}, a(255)=480, a recurrent term;
n=247: list={247,96,72,96,...}, a(247)=247, a transient term, here the initial value.

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

Cf. A000010, A000203, A062401, A062402, A095955, A096859, A096865.

Cf. also A096864.

gf[x_] :=DivisorSigma[1, EulerPhi[x]] itef[x_, len_] :=NestList[fs, x, len] Table[Max[itef[w, 20]], {w, 1, 256}]

(define (A096861 n) (let loop ((visited (list n)) (m n)) (let ((next (A062401 (car visited)))) (cond ((member next visited) m) (else (loop (cons next visited) (max 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

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

Labos Elemer, Jul 16 2004

nonn

			All odd and certain even integers belong here.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

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

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 *)

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}]

cp's OEIS Frontend

A096852 a(n) is the length of terminal cycle of the trajectory of f(x)=phi(sigma(x)) if started at 2^n.

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

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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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A096859 Function A062401(x) = phi(sigma(x)) = f(x) is iterated. Starting with n, a(n) is the count of distinct terms arising in trajectory; a(n)=t(n)+c(n)=t+c, where t=number of transient terms, c=number of recurrent terms (in the terminal cycle).

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A096865 Function A062401(x) = phi(sigma(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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A096861 Function A062401(x) = phi(sigma(x)) = f(x) is iterated. Starting with n, a(n) is the largest term arising in trajectory.

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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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