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A006517 Numbers n such that n divides 2^n + 2.
(Formerly M1719)
5
1, 2, 6, 66, 946, 8646, 180246, 199606, 265826, 383846, 1234806, 3757426, 9880278, 14304466, 23612226, 27052806, 43091686, 63265474, 66154726, 69410706, 81517766, 106047766, 129773526, 130520566, 149497986, 184416166, 279383126 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

All terms greater than 1 are even. If an odd term n>1 exists then n = m*2^k + 1 for some k>=1 and odd m. Then n divides 2^(m*2^k) + 1 and so does every prime factor p of n, implying that 2^(k+1) divides the multiplicative order of 2 modulo p and thus p-1. Therefore n = m*2^k + 1 is the product of prime factors of the form t*2^(k+1) + 1, implying that n-1 is divisible by 2^(k+1), a contradiction. - Max Alekseyev, Mar 16 2009

The sequence is infinite. In fact, its intersection with A055685 (given by A219037) is infinite (see Li et al. link). - Max Alekseyev, Oct 11 2012

All terms greater than 6 have at least three distinct prime factors. - Robert Israel, Aug 21 2014

REFERENCES

R. Honsberger, Mathematical Gems, M.A.A., 1973, p. 142.

SierpiƄski, W. 250 Problems in Elementary Number Theory. New York: American Elsevier, 1970. Problem #18

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

LINKS

Table of n, a(n) for n=1..27.

Kin Y. Li et al., Solution to Problem 323, Mathematical Excalibur 14(2), 2009, p. 3.

MATHEMATICA

Do[ If[ PowerMod[ 2, n, n ] + 2 == n, Print[n]], {n, 2, 1500000000, 4} ]

PROG

(PARI) e323(n) = {for (i=1, n, if ((2^i+2) % i == 0, print1(i, ", ")); ); } \\ Michel Marcus, Oct 07 2012

(PARI) is_A006517(n)=!(Mod(2, n)^n+2)  \\ M. F. Hasler, Oct 08 2012

CROSSREFS

Cf. A006521.

Sequence in context: A082619 A046399 A082617 * A217630 A091458 A167006

Adjacent sequences:  A006514 A006515 A006516 * A006518 A006519 A006520

KEYWORD

nonn,nice

AUTHOR

N. J. A. Sloane, David W. Wilson

EXTENSIONS

Corrected and extended by Joe K. Crump (joecr(AT)carolina.rr.com), Sep 12 2000 and Robert G. Wilson v, Sep 13 2000

STATUS

approved

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Last modified May 25 19:28 EDT 2017. Contains 287059 sequences.