A096852 -id:A096852 - cp's OEIS frontend

A096857 a[n] is the length of terminal cycle of the trajectory of g[x]=sigma(phi(x)) if started at 2^n. Formally identical to A096852, but arguments are shifted by 1 and the iterated functions are different!.

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: 1

Labos Elemer, Jul 19 2004

nonn

Offset=1 in contrast to A096852, where offset=0. Also the iterated functions deviate: A062401 iterated in A096852 and A062402 is repeated here; A096852(n)=A096857(n+1) appears to be true. While cycle-lengths seem identical, the composition of cycles are mostly different!

			n=5:iv=32 list={32,[31,72,60]} length=a(5)=3, while the parallel case of A096852(n)=b(n) is b[4] with [16,24,30] cycle.
Also A096857[11] starts with 2048 ends in 6-cycle: {2048,2047,4123,10890,8928,[9906,9920,12264,10200,6138,6045],9906,..
while A096852[11-1]=6 and the relevant 6-cycle is {1024,1936,3240,2640,[2880,3024,3840,3456,2560,1800],2880,... These are different cycles with identical lengths.
The initial value 146 leads to list with enormous terms.

Cf. A062401, A062402, A096852, A096858.

f[n_] := DivisorSigma[1, EulerPhi[n]]; g[n_] := Block[{l = NestWhileList[f, 2^n, UnsameQ, All]}, -Subtract @@ Flatten[Position[l, l[[ -1]]]]]; Table[ g[n], {n, 25}] (* Robert G. Wilson v, Jul 21 2004 *)

A096994 Number of transient terms if f(x)=phi(sigma(x))=A062401 is iterated at initial value 2^n. Equilibrium terms are listed in A096852.

Original entry on oeis.org

0, 0, 0, 0, 0, 2, 2, 0, 1, 2, 4, 1, 2, 5, 14, 0, 5, 7, 2, 14, 8, 3, 64, 43, 81, 82, 76, 74, 47, 25, 42, 0

Offset: 0

Labos Elemer, Jul 22 2004

For transient lengths for iterations of A062401(x) or A062402(x) if started at 2^n, A096994(n) + 1 = A096995(n). Corresponding cycle lengths satisfy A096852(n-1) = A096857(n). Behind these observations several relationships stand, e.g., sigma(A062401(x)) = A062402(sigma(x)) or phi(A062402(x)) = A062401(phi(x)).

			n=0: trajectory = {1,1,..} so a(0)=0;
n=14: transient-length=14, cycle-length=2, a(14)=14, A096852(14)=2; trajectory ={16384, 27000, 23040, 21600, 17280, 15360, 15488, 13824, 9600, 7680, 7200, 12960, 11880, 11520, [10368,14080], 10368, ...}.
Values of a(n) for n > 31, with -1 signifying transient lengths yet unknown after 10^4 iterations of f(x): -1, 7, 51, 70, 23, 39, 11, -1, 37, 107, 30, -1, 145, 25, 21, 36, -1, -1, -1, -1, 31, -1, 452, -1, 449, 447, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, 40, -1, -1, -1, -1, -1, -1, -1, 279, -1. - _Michael De Vlieger_, May 15 2017

Cf. A096852, A062401.

With[{nn = 10^3}, Table[Count[Values@ PositionIndex@ #, k_ /; Length@ k == 1] &@ NestList[EulerPhi@ DivisorSigma[1, #] &, 2^n, nn] /. k_ /; k == nn + 1 -> -1, {n, 31}] ] (* Michael De Vlieger, May 15 2017, Version 10 *)

A062401 a(n) = phi(sigma(n)).

Original entry on oeis.org

1, 2, 2, 6, 2, 4, 4, 8, 12, 6, 4, 12, 6, 8, 8, 30, 6, 24, 8, 12, 16, 12, 8, 16, 30, 12, 16, 24, 8, 24, 16, 36, 16, 18, 16, 72, 18, 16, 24, 24, 12, 32, 20, 24, 24, 24, 16, 60, 36, 60, 24, 42, 18, 32, 24, 32, 32, 24, 16, 48, 30, 32, 48, 126, 24, 48, 32, 36, 32, 48, 24, 96, 36, 36, 60

Offset: 1

Jason Earls, Jul 08 2001

nonn

			a(9) = 12 because sigma(9) = 13 and phi(13) = 12.

D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, p. 14.

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A000203, A000010, A062402, A096852, A096857, A096994, A096995, A033632.

Cf. A001229 (fixed points).

a062401 = a000010 . a000203  -- Reinhard Zumkeller, Jan 04 2013

with(numtheory); A062401:=n->phi(sigma(n)); seq(A062401(n), n=1..50); # Wesley Ivan Hurt, Apr 07 2014

Table[EulerPhi[DivisorSigma[1, n]], {n, 1, 80}] (* Carl Najafi, Aug 16 2011 *)

PARI
```
vector(150, n, eulerphi(sigma(n)))
```

for (n=1, 10000, write("b062401.txt", n, " ", eulerphi(sigma(n))) ) \\ Harry J. Smith, Aug 07 2009

sigma(a(n)) = A062402(sigma(n)) or phi(A062402(n)) = a(phi(n)). - Labos Elemer, Jul 22 2004

A062402 a(n) = sigma(phi(n)).

Original entry on oeis.org

1, 1, 3, 3, 7, 3, 12, 7, 12, 7, 18, 7, 28, 12, 15, 15, 31, 12, 39, 15, 28, 18, 36, 15, 42, 28, 39, 28, 56, 15, 72, 31, 42, 31, 60, 28, 91, 39, 60, 31, 90, 28, 96, 42, 60, 36, 72, 31, 96, 42, 63, 60, 98, 39, 90, 60, 91, 56, 90, 31, 168, 72, 91, 63, 124, 42, 144, 63, 84, 60, 144

Offset: 1

Jason Earls, Jul 08 2001

nonn

Makowski and Schinzel conjectured in 1964 that a(n) = sigma(phi(n)) >= n/2 for all n (B42 of Guy Unsolved Problems...). This has been verified for various classes of numbers and proved to be true in general if it is true for squarefree integers (see Cohen's paper). - Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Sep 07 2002

Atanassov proves the above conjecture. - Charles R Greathouse IV, Dec 06 2016

			a(9)= 12 because phi(9)= 6 and sigma(6)= 12.

Krassimir T. Atanassov, One property of φ and σ functions, Bull. Number Theory Related Topics 13 (1989), pp. 29-37.
A. Makowski and A. Schinzel, On the functions phi(n) and sigma(n), Colloq. Math. 13, 95-99 (1964).
D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, p. 13.

T. D. Noe, Table of n, a(n) for n = 1..10000
G. L. Cohen, On a conjecture of Makowski and Schinzel. Colloq. Math. 74, No. 1, 1-8 (1997).
A. Grytczuk, F. Luca and M. Wojtowicz, On a conjecture of Makowski and Schinzel concerning the composition of the arithmetic functions sigma and phi, Colloq. Math. 86, No. 1, 31-36 (2000).
F. Luca and C. Pomerance, On some problems of Makowski-Schinzel and Erdos concerning the arithmetical functions phi and sigma, Colloq. Math. 92, No. 1, 111-130 (2002).

Cf. A000203, A000010, A062401, A096852, A096857, A096994, A096995, A033632.

a062402 = a000203 . a000010  -- Reinhard Zumkeller, Jan 04 2013

[SumOfDivisors(EulerPhi(n)): n in [1..100]] //  Marius A. Burtea, Jan 19 2019

with(numtheory); A062402:=n->sigma(phi(n)); seq(A062402(k), k=1..100); # Wesley Ivan Hurt, Nov 01 2013

Table[DivisorSigma[1, EulerPhi[n]], {n, 1, 80}] (* Carl Najafi, Aug 16 2011 *)

a(n)=sigma(eulerphi(n));
vector(150,n,a(n))

from sympy import divisor_sigma, totient
print([divisor_sigma(totient(n)) for n in range(1, 101)]) # Indranil Ghosh, Mar 18 2017

sigma(A062401(x)) = a(sigma(x)) or phi(a(x)) = A062401(phi(x)). - Labos Elemer, Jul 22 2004

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

A097004 Function A062402(x)=phi(sigma(x)) is iterated. Starting with 2^n, the n-th power of 2, a(n) is the count of distinct terms arising in 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, 2, 3, 2, 4, 5, 5, 2, 4, 5, 11, 4, 4, 12, 17, 2, 8, 11, 14, 26, 11, 6, 80, 59, 100, 101, 95, 93, 60, 38, 55, 2

Offset: 0

Labos Elemer, Jul 21 2004

			n=13: 2^n=8192, trajectory ={8192,8191,26208,[20440],.. }, a[13]=3+1=4 with 3 transients and one recurrent term.

Cf. A000010, A000203, A062402, A096852, A096857.

EulerPhi[DivisorSigma[1, x]] itef[x_, len_] :=NestList[fs, x, len] Table[Length[Union[itef[2^w, 1000]]], {w, 1, 256}]

A096999 A062401(x)=phi[sigma(x)] function is iterated; initial value=2^n; a(n)=largest term of trajectory.

Original entry on oeis.org

2, 6, 8, 30, 96, 126, 128, 480, 600, 3840, 2048, 8190, 10584, 27000, 32768, 196560, 311040, 851840, 1161216, 2250600, 2640704, 150992640, 150992640, 283740364800, 283740364800, 283740364800, 283740364800, 7608287232, 7608287232

Offset: 1

Labos Elemer, Jul 21 2004

nonn

			n=8: 2^n=256, list={256,432,480,432,...}, max=a(8)=480

Cf. A000010, A000203, A062401, A096852.

fs[x_] :=EulerPhi[DivisorSigma[1, x]] itef[x_, hos_] :=NestList[fs, x, hos] Table[Max[itef[2^w, 200]], {w, 1, 30}]

A097000 A062401(x)=phi[sigma(x)] function is iterated; initial value=2^n; a(n)=smallest term of trajectory.

Original entry on oeis.org

2, 4, 8, 16, 32, 64, 128, 256, 432, 1024, 1728, 4096, 1800, 7200, 32768, 65536, 131072, 262144, 326592, 1036800, 1658880, 4194304, 4838400, 16777216, 33554432, 67108864, 82301184, 207360000, 361267200, 414720000

Offset: 1

Labos Elemer, Jul 21 2004

nonn

			n=8: 2^n=256, list={256,432,480,432,...}, max=a(8)=256, the start value;
n=13:2^n=8192,list={8192,10584,8640,8064,6144,3456,[2560,1800,2880, 3024,3840,3456],2560,..}, min=1800=a(13), a term of 6-cycle.

Cf. A000010, A000203, A062401, A096852.

fs[x_] :=EulerPhi[DivisorSigma[1, x]] itef[x_, hos_] :=NestList[fs, x, hos] Table[Min[itef[2^w, 200]], {w, 1, 30}]

A097001 A062402(x)=sigma(phi[x]) function is iterated; initial value=2^n; a(n)=largest term of trajectory.

Original entry on oeis.org

2, 4, 12, 16, 72, 252, 312, 256, 1512, 1860, 12264, 6552, 26208, 34200, 93600, 65536, 833280, 1116024, 2239920, 4464096, 9865440, 8124480, 569540160, 569540160, 1100946774480, 1100946774480, 1100946774480, 1100946774480, 34696672920

Offset: 1

Labos Elemer, Jul 21 2004

nonn

			n=13: 2^n=8192, list={8192,8191,26208,[20440],20440,.. a[13]=26208 arose in transient.

Cf. A000010, A000203, A062402, A096852.

gf[x_] :=DivisorSigma[1, EulerPhi[x]] gite[x_, hos_] :=NestList[gf, x, hos] Table[Max[gite[2^w, 200]], {w, 1, 30}]

A097003 Function A062402[x]=phi[sigma[x]] is iterated. a(n) is the number of distinct terms arising in the trajectory of 2^n; 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, 1, 2, 1, 3, 4, 4, 1, 3, 4, 10, 3, 3, 11, 16, 1, 7, 10, 13, 25, 10, 5, 79, 58, 99, 100, 94, 92, 59, 37, 54, 1

Offset: 0

Labos Elemer, Jul 21 2004

nonn

Concerning this sequence and A097004, A096994, A096995: in all 4 cases the initial value is 2^n and a certain function is iterated. They differ either in the function or in what is computed for that iteration.
Glossary: t+c = total count of transient+cycle terms, t = count of transient terms
Sequence 1: A062401 is iterated t+c is computed => this sequence
Sequence 2: A062402 is iterated t+c is computed => A097004
Sequence 3: A062401 is iterated t is computed => A096994
Sequence 4: A062402 is iterated t is computed => A096995

			n=13: 2^n=8192, trajectory ={8192, 10584, 8640, 8064, 6144, [3456, 2560, 1800, 2880, 3024, 3840], 3456, 2560, ..}, t+c=a(13)=5+6=11;

Cf. A000010, A000203, A062402, A096852.

EulerPhi[DivisorSigma[1, x]] itef[x_, len_] :=NestList[fs, x, len] Table[Length[Union[itef[2^w, 20]]], {w, 1, 256}]

cp's OEIS Frontend

A096857 a[n] is the length of terminal cycle of the trajectory of g[x]=sigma(phi(x)) if started at 2^n. Formally identical to A096852, but arguments are shifted by 1 and the iterated functions are different!.

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A096994 Number of transient terms if f(x)=phi(sigma(x))=A062401 is iterated at initial value 2^n. Equilibrium terms are listed in A096852.

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A062401 a(n) = phi(sigma(n)).

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A062402 a(n) = sigma(phi(n)).

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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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A097004 Function A062402(x)=phi(sigma(x)) is iterated. Starting with 2^n, the n-th power of 2, a(n) is the count of distinct terms arising in 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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A096999 A062401(x)=phi[sigma(x)] function is iterated; initial value=2^n; a(n)=largest term of trajectory.

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A097000 A062401(x)=phi[sigma(x)] function is iterated; initial value=2^n; a(n)=smallest term of trajectory.

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