A096994 -id:A096994 - cp's OEIS frontend

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

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

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}]

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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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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].

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