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A007694 Numbers n such that phi(n) divides n.
(Formerly M0992)
18
1, 2, 4, 6, 8, 12, 16, 18, 24, 32, 36, 48, 54, 64, 72, 96, 108, 128, 144, 162, 192, 216, 256, 288, 324, 384, 432, 486, 512, 576, 648, 768, 864, 972, 1024, 1152, 1296, 1458, 1536, 1728, 1944, 2048, 2304, 2592, 2916, 3072, 3456, 3888, 4096 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

a(n) divides p^a(n) - 1 for all primes p >= 5. - Benoit Cloitre, Mar 22 2002

Also n such that sum( d divides n, mu(d)/d) has numerator of 1. - Benoit Cloitre, Apr 15 2002

n is here if and only if phi(n) divides also cototient(n). On the other hand, cototient(n) divides phi(n) if and only if n is a prime or power of a prime. - Labos Elemer, May 03 2002

It follows that n/phi(n) = 2 if n is a power of 2 and equal to 3 if n is of the form 6*A003586. - Gary Detlefs, Jun 28 2011

1 and even 3-smooth numbers, cf. A003586.

REFERENCES

J.-M. De Koninck & A. Mercier, 1001 Problemes en Theorie Classique Des Nombres, Problem 526 pp. 71; 256, Ellipses Paris 2004.

Problem E3037, Amer. Math. Monthly 93 (1986), 656-657.

Sarkozy A. and Suranyi J., Number Theory Problem Book (in Hungarian), Tankonyvkiado, Budapest, 1972.

W. Sierpinski, Elementary Theory of Numbers, Warsaw, 1964.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

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

W. Sierpi\'{n}ski, Elementary Theory of Numbers, Warszawa 1964.

FORMULA

n/phi(n) is an integer if and only if n = 1 or n = 2^w * 3^u for w > 0 and u >= 0.

EXAMPLE

12 is in the sequence because 12/phi(12) = 12/4 = 3, which is an integer.

16 is in the sequence because 16/phi(16) = 16/8 = 2, which is an integer.

20 is not in the sequence because 20/phi(20) = 20/8 = 5/2 = 2.5, which is not an integer.

MATHEMATICA

Select[ Range[5000], IntegerQ[ #/EulerPhi[ # ]] &]

PROG

(R) library(numbers); j=N=1

while(j<200) if(isNatural((N=N+1)/eulersPhi(N))) dtot[(j=j+1)]=N # Christian N. K. Anderson, Apr 04 2013

(PARI) for(n=1, 10^6, if (n%eulerphi(n)==0, print1(n, ", "))); /* Joerg Arndt, Apr 04 2013 */

(Haskell)

a007694 n = a007694_list !! (n-1)

a007694_list = 1 : filter even a003586_list

-- Reinhard Zumkeller, Jan 06 2014

CROSSREFS

Cf. A000010, A049237, A007694, A007947, A003557, A023200.

Cf. A003586.

Cf. A033950, A235353 (subsequence).

Sequence in context: A227270 A145853 A064527 * A219653 A050622 A082662

Adjacent sequences:  A007691 A007692 A007693 * A007695 A007696 A007697

KEYWORD

nonn,nice,easy

AUTHOR

N. J. A. Sloane, Robert G. Wilson v

STATUS

approved

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Last modified October 1 05:22 EDT 2014. Contains 247503 sequences.