A062401 -id:A062401 - cp's OEIS frontend

A096855 a(n) = A062401(2^n + 1).

2, 2, 2, 12, 6, 16, 24, 80, 84, 320, 360, 864, 1320, 5456, 5184, 15744, 19800, 52800, 69120, 349520, 370080, 1036800, 1425600, 3640896, 4741632, 13989888, 27091584, 76743040, 94656000, 166387200, 412473600, 1407389952, 1420488192, 3459760128, 6502788864, 14778408960

Offset: 0

Labos Elemer, Jul 19 2004

nonn

Amiram Eldar, Table of n, a(n) for n = 0..400

Cf. A000010, A062401, A062402, A053287, A096853, A069061, A096854, A096855, A096856.

Table[ EulerPhi[ DivisorSigma[1, 2^n+1]], {n, 0, 33}]

a(n) = A000010(A069061(n)). - Amiram Eldar, Jun 04 2024

Edited and extended by Robert G. Wilson v, Jul 23 2004

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

A096860 Function A062401(x) = phi(sigma(x)) = f(x) is iterated. Starting with n, a(n) is the count of distinct terms arising in the transient of this trajectory, that is: a(n) = A096859(n) - A095955(n).

Original entry on oeis.org

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

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=4;
n=244: list={244,180,144,360,288,[432,480],432,...}, a(244)=4.
a(n)=0 means that n is a recurrent term from A096850.

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

Cf. A000010, A000203, A062401, A062402, A095955, A096859, A096861, A096865.

Cf. also A096863.

With[{nn = 120}, Array[Length@ Union@ # - Length@ Select[Tally@ #, Last@ # > 1 &] &@ NestList[EulerPhi@ DivisorSigma[1, #] &, #, nn] &, 105]] (* Michael De Vlieger, Nov 18 2017 *)

(define (A096860 n) (let loop ((visited (list n))) (let ((next (A062401 (car visited)))) (cond ((member next visited) => (lambda (transientplusone) (- (length transientplusone) 1))) (else (loop (cons next visited))))))) ;; Antti Karttunen, Nov 18 2017

A295310 a(n) = gcd(n, A062401(n)), where A062401(n) = phi(sigma(n)).

Original entry on oeis.org

1, 2, 1, 2, 1, 2, 1, 8, 3, 2, 1, 12, 1, 2, 1, 2, 1, 6, 1, 4, 1, 2, 1, 8, 5, 2, 1, 4, 1, 6, 1, 4, 1, 2, 1, 36, 1, 2, 3, 8, 1, 2, 1, 4, 3, 2, 1, 12, 1, 10, 3, 2, 1, 2, 1, 8, 1, 2, 1, 12, 1, 2, 3, 2, 1, 6, 1, 4, 1, 2, 1, 24, 1, 2, 15, 4, 1, 6, 1, 20, 1, 2, 1, 12, 1, 2, 1, 8, 1, 18, 1, 4, 1, 2, 1, 24, 1, 2, 3, 20, 1, 6, 1, 8, 1

Offset: 1

Antti Karttunen, Nov 22 2017

nonn

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

Cf. A000010, A000203, A062401, A295301, A295311, A295312, A295313.

Array[GCD[#, EulerPhi[DivisorSigma[1, #]]] &, 105] (* Michael De Vlieger, Nov 23 2017 *)

a(n) = gcd(n, eulerphi(sigma(n))); \\ Michel Marcus, Nov 23 2017

a(n) = gcd(n, A000010(A000203(n))).

A295312 a(n) = A062401(n) / A295310(n) = phi(sigma(n)) / gcd(n, phi(sigma(n))).

Original entry on oeis.org

1, 1, 2, 3, 2, 2, 4, 1, 4, 3, 4, 1, 6, 4, 8, 15, 6, 4, 8, 3, 16, 6, 8, 2, 6, 6, 16, 6, 8, 4, 16, 9, 16, 9, 16, 2, 18, 8, 8, 3, 12, 16, 20, 6, 8, 12, 16, 5, 36, 6, 8, 21, 18, 16, 24, 4, 32, 12, 16, 4, 30, 16, 16, 63, 24, 8, 32, 9, 32, 24, 24, 4, 36, 18, 4, 12, 32, 8, 32, 3, 110, 18, 24, 8, 36, 20, 32, 6, 24, 4, 48, 12, 64, 24, 32, 3

Offset: 1

Antti Karttunen, Nov 22 2017

nonn

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

Cf. A000010, A000203, A062401, A295301, A295310, A295311, A295315.

a(n) = my(fsn = eulerphi(sigma(n))); fsn/gcd(n, fsn); \\ Michel Marcus, Nov 23 2017

a(n) = A000010(A000203(n)) / gcd(n, A000010(A000203(n))).

A353647 Möbius transform of A062401, where A062401(n) = phi(sigma(n)).

Original entry on oeis.org

1, 1, 1, 4, 1, 1, 3, 2, 10, 3, 3, 4, 5, 3, 5, 22, 5, 10, 7, 2, 11, 7, 7, 2, 28, 5, 4, 12, 7, 11, 15, 6, 11, 11, 11, 40, 17, 7, 17, 10, 11, 11, 19, 8, 6, 15, 15, 22, 32, 26, 17, 26, 17, 4, 19, 6, 23, 15, 15, 14, 29, 15, 22, 90, 17, 23, 31, 14, 23, 25, 23, 20, 35, 17, 24, 28, 25, 17, 31, 14, 94, 23, 23, 44, 29, 19

Offset: 1

Antti Karttunen, May 06 2022

sign

The first negative term is a(1944) = -76.

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

Cf. A000010, A000203, A008683, A062401.

a[n_] := DivisorSum[n, MoebiusMu[n/#] * EulerPhi[DivisorSigma[1, #]] &]; Array[a, 100] (* Amiram Eldar, May 06 2022 *)

A353647(n) = sumdiv(n,d,moebius(n/d)*eulerphi(sigma(d)));

a(n) = Sum_{d|n} A008683(n/d) * A062401(d).

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

A097005 A062401(x) is iterated. Initial value = n!. a(n) is the path-length of trajectory = count of transients and recurrent terms, i.e., all distinct states arising in trajectory.

Original entry on oeis.org

1, 1, 1, 2, 3, 3, 1, 9, 6, 14, 2, 51, 35, 81, 32, 31

Offset: 0

Labos Elemer, Jul 22 2004

			n=10: 10! = 3628800; trajectory ={3628800, 5702400, 5702400, ...} a(10) = 2, one transient and 1 cycle term.

Cf. A000010, A000142, A000203, A062401, A096859.

fs[x_]:=EulerPhi[DivisorSigma[1,x]] itef[x_,len_]:=NestList[fs,x,len] Table[Length[Union[itef[w!,1000]]],{w,0,16}]

a(n) = A096859(n!) = A096859(A000142(n)). - Michel Marcus, Jul 27 2017

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}]

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

cp's OEIS Frontend

A096855 a(n) = A062401(2^n + 1).

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A096860 Function A062401(x) = phi(sigma(x)) = f(x) is iterated. Starting with n, a(n) is the count of distinct terms arising in the transient of this trajectory, that is: a(n) = A096859(n) - A095955(n).

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A295310 a(n) = gcd(n, A062401(n)), where A062401(n) = phi(sigma(n)).

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A295312 a(n) = A062401(n) / A295310(n) = phi(sigma(n)) / gcd(n, phi(sigma(n))).

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A353647 Möbius transform of A062401, where A062401(n) = phi(sigma(n)).

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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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A097005 A062401(x) is iterated. Initial value = n!. a(n) is the path-length of trajectory = count of transients and recurrent terms, i.e., all distinct states arising in trajectory.

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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.