

A006517


Numbers k such that k divides 2^k + 2.
(Formerly M1719)


20



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
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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 p1. Therefore n = m*2^k + 1 is the product of prime factors of the form t*2^(k+1) + 1, implying that n1 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.
SierpiĆski, 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

Giovanni Resta, Table of n, a(n) for n = 1..150
Kin Y. Li et al., Solution to Problem 323, Mathematical Excalibur 14(2), 2009, p. 3.
V. Meally, Letter to N. J. A. Sloane, May 1975


MATHEMATICA

Do[ If[ PowerMod[ 2, n, n ] + 2 == n, Print[n]], {n, 2, 1500000000, 4} ]
Join[{1}, Select[Range[28*10^7], PowerMod[2, #, #]==#2&]] (* Harvey P. Dale, Aug 13 2018 *)


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, A015888, A015889, A015891, A015892, A015893, A015897, A015898, A015902, A015903, A015904, A015905, A015906.
Sequence in context: A082619 A046399 A082617 * A217630 A091458 A335934
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



