A062402 -id:A062402 - cp's OEIS frontend

A097002 A062402(x)=sigma(phi[x]) function is iterated; initial value=2^n; a(n)=smallest term of trajectory.

1, 3, 7, 15, 31, 63, 127, 255, 511, 1023, 2047, 4095, 8191, 6045, 24552, 65535, 131071, 262143, 524287, 1048575, 2097151, 4194303, 8388607, 16777215, 33554431, 67108863, 134217727, 268435455, 536870911, 1073741823

Offset: 1

Labos Elemer, Jul 21 2004

nonn

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

Cf. A000010, A000203, A062402, A096857.

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

A097006 Consider the function f(x)=sigma(phi(x))=A062402(x) iterated with initial value n!; a(n) is the path-length of trajectory.

Original entry on oeis.org

1, 1, 2, 2, 2, 5, 6, 5, 10, 10, 17, 49, 91

Offset: 0

Labos Elemer, Jul 22 2004

The path length is the total number of transient and recurrent terms.

After 12000 iterations, f(13!) reaches 583880633503221176888439640142607059743547704176558111997560422400000.

			n=10: 10!=3628800; the trajectory is 3628800, 2972970, 2221560, 1915992, 1768767, 2877420, [1965840, 2227680, 1310680, 1591200, 1277874, 1307124, 1110488, 2010960, 1488032, 1981496, 2239920], [1965840, ...], ...; thus a(10)=17, with 6 transient and 11 recurrent states.

Cf. A000203, A000010, A062401, A000142, A097005.

f[n_] := DivisorSigma[1, EulerPhi[n]]; g[n_] := Length[ NestWhileList[ f, n, UnsameQ, All]] - 1; Table[ g[n! ], {n, 12}] (* Robert G. Wilson v, Jul 23 2004 *)

Edited by Robert G. Wilson v, Jul 23 2004

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

A033632 Numbers k such that sigma(phi(k)) = phi(sigma(k)).

Original entry on oeis.org

1, 9, 225, 242, 516, 729, 3872, 13932, 14406, 17672, 18225, 20124, 21780, 29262, 29616, 45996, 65025, 76832, 92778, 95916, 106092, 106308, 114630, 114930, 121872, 125652, 140130, 140625, 145794, 149124, 160986, 179562, 185100, 234876, 248652, 252978, 256860

Offset: 1

Jud McCranie

The largest term of this sequence that I found is 3^9550. Also, if (1/2)*(3^(k+1)-1) is prime (k+1 is a term of A028491) then 3^k is in the sequence, namely sigma(phi(3^k)) = phi(sigma(3^k)) (the proof is easy). - Farideh Firoozbakht, Feb 09 2005

R. K. Guy, Unsolved Problems in Number Theory, 2nd edition, Springer Verlag, 1994, section B42, p. 99.

Donovan Johnson, Table of n, a(n) for n = 1..10000 (first 200 terms from T. D. Noe)
S. W. Golomb, Equality among number-theoretic functions, Unpublished manuscript. (Annotated scanned copy)
Walter Nissen, sigma(phi(n)) = phi(sigma(n)), Up for the Count !
Walter Nissen, sigma(phi(n)) = phi(sigma(n)): From "5" to "5 figures", Up for the Count !, Nov. 14, 2000
Walter Nissen, Addendum to : sigma(phi()): From "5" to "5 figures", Up for the Count !, June 8, 2008
Walter Nissen, Elaboration of : sigma(phi()): From "5" to "5 figures", Up for the Count !, Oct. 15, 2010

Cf. A000203, A000010, A028491, A078148.

a033632 n = a033632_list !! (n-1)
a033632_list = filter (\x -> a062401 x == a062402 x) [1..]
-- Reinhard Zumkeller, Jan 04 2013

Select[ Range[ 10^6 ], DivisorSigma[ 1, EulerPhi[ # ] ] == EulerPhi[ DivisorSigma[ 1, # ] ] & ]

is(n)=sigma(eulerphi(n))==eulerphi(sigma(n)) \\ Charles R Greathouse IV, May 09 2013

from sympy import divisor_sigma as sigma, totient as phi
def ok(n): return sigma(phi(n)) == phi(sigma(n))
def aupto(nn): return [m for m in range(1, nn+1) if ok(m)]
print(aupto(10**4)) # Michael S. Branicky, Jan 09 2021

A062401(a(n)) = A062402(a(n)). - Reinhard Zumkeller, Jan 04 2013

A065395 Commutator of sigma and phi functions.

Original entry on oeis.org

0, -1, 1, -3, 5, -1, 8, -1, 0, 1, 14, -5, 22, 4, 7, -15, 25, -12, 31, 3, 12, 6, 28, -1, 12, 16, 23, 4, 48, -9, 56, -5, 26, 13, 44, -44, 73, 23, 36, 7, 78, -4, 76, 18, 36, 12, 56, -29, 60, -18, 39, 18, 80, 7, 66, 28, 59, 32, 74, -17, 138, 40, 43, -63, 100, -6

Offset: 1

Labos Elemer, Nov 05 2001

Golomb (1993) proved that the terms are both positive and negative infinitely often. - Amiram Eldar, Feb 27 2024

			n = 13: sigma(13) = 14, phi(14) = 6, phi(13) = 12, sigma(12) = 28, a(13) = 28-6 = 22.

Solomon W. Golomb, Equality among number-theoretic functions, Abstracts Amer. Math. Soc., Vol. 14 (1993), pp. 415-416.

Harry J. Smith, Table of n, a(n) for n = 1..1000
Jean-Marie De Koninck and Florian Luca, On the composition of the Euler function and the sum of divisors function, Colloquium Mathematicum, Vol. 108, No. 1 (2007), pp. 31-51.
Solomon W. Golomb, Equality among number-theoretic functions, Unpublished manuscript. (Annotated scanned copy)

Cf. A000010, A000203, A033632 (positions of 0's), A062401, A062402.

[DivisorSigma(1, EulerPhi(n))-EulerPhi(DivisorSigma(1, n)): n in [1..70]]; // Bruno Berselli, Oct 20 2015

with(numtheory); A065395:=n->sigma(phi(n))-phi(sigma(n)); seq(A065395(n), n=1..100); # Wesley Ivan Hurt, Dec 26 2013

Table[DivisorSigma[1, EulerPhi[n]] - EulerPhi[DivisorSigma[1, n]], {n, 100}] (* T. D. Noe, Nov 04 2013 *)

a(n) = { sigma(eulerphi(n)) - eulerphi(sigma(n)) } \\ Harry J. Smith, Oct 18 2009

a(n) = sigma(phi(n)) - phi(sigma(n)) = A000203(A000010(n)) - A000010(A000203(n)).

a(n) = A062402(n) - A062401(n). - Amiram Eldar, Feb 27 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

A062514 Numbers k such that sigma(phi(k)) is a prime.

Original entry on oeis.org

3, 4, 5, 6, 8, 10, 12, 17, 32, 34, 40, 48, 60, 85, 128, 136, 160, 170, 192, 204, 240, 4369, 8192, 8224, 8704, 8738, 10240, 10280, 10880, 12288, 12336, 13056, 15360, 15420, 16320, 65537, 131072, 131074, 131584, 139264, 139808, 163840, 164480, 174080

Offset: 1

Jason Earls, Jul 09 2001

Numbers k such that A062402(k) is prime.

			Phi(174080)=65536. Sigma(65536) = 131071, a prime.

Michel Marcus, Table of n, a(n) for n = 1..105 (terms 1..73 from Harry J. Smith)

Cf. A000010, A000203, A062402.

je=[]; for(n=1,300000,s=sigma(eulerphi(n)); if(isprime(s),je=concat(je,n))); je

{ n=0; for (m=1, 10^9, if(isprime(sigma(eulerphi(m))), write("b062514.txt", n++, " ", m); if (n==73, break)) ) } \\ Harry J. Smith, Aug 08 2009

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!.

Original entry on oeis.org

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 *)

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

cp's OEIS Frontend

A097002 A062402(x)=sigma(phi[x]) function is iterated; initial value=2^n; a(n)=smallest term of trajectory.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

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].

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A097006 Consider the function f(x)=sigma(phi(x))=A062402(x) iterated with initial value n!; a(n) is the path-length of trajectory.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Extensions

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

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

PARI

Formula

A033632 Numbers k such that sigma(phi(k)) = phi(sigma(k)).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Python

Formula

A065395 Commutator of sigma and phi functions.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

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).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs