A096994 -id:A096994

Search: a096994 -id:a096994

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A062401

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

+10
106

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

REFERENCES

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

LINKS

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

FORMULA

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

EXAMPLE

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

MAPLE

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

MATHEMATICA

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

PROG

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

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

(Haskell)

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

CROSSREFS

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

Cf. A001229 (fixed points).

KEYWORD

nonn

AUTHOR

Jason Earls, Jul 08 2001

STATUS

approved

A062402

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

+10
75

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

COMMENTS

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

REFERENCES

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.

LINKS

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).

FORMULA

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

EXAMPLE

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

MAPLE

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

MATHEMATICA

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

PROG

(PARI) a(n)=sigma(eulerphi(n));

vector(150, n, a(n))

(Haskell)

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

(Python)

from sympy import divisor_sigma, totient

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

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

CROSSREFS

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

KEYWORD

nonn

AUTHOR

Jason Earls, Jul 08 2001

STATUS

approved

A096995

Number of transient terms if f(x) = sigma(phi(x)) = A062402 is iterated at initial value = 2^n.

+10
4

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,7

COMMENTS

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

LINKS

Table of n, a(n) for n=0..32.

EXAMPLE

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.

MATHEMATICA

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 *)

CROSSREFS

Cf. A062402, A062401, A096857, A096852, A096994.

KEYWORD

nonn,more

AUTHOR

Labos Elemer, Jul 22 2004

EXTENSIONS

Edited and corrected by Klaus Brockhaus, Jul 19 2007

STATUS

approved

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].

+10
0

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

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

LINKS

Table of n, a(n) for n=0..31.

EXAMPLE

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;

MATHEMATICA

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

CROSSREFS

Cf. A000010, A000203, A062402, A096852.

KEYWORD

nonn

AUTHOR

Labos Elemer, Jul 21 2004

STATUS

approved

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