1,2

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 withA055685(given byA219037) 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

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

W. Sierpiński, 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).

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

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 *)

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

Cf.A006521,A015888,A015889,A015891,A015892,A015893,A015897,A015898,A015902,A015903,A015904,A015905,A015906.

Sequence in context:A082619A046399A082617*A217630A091458A335934

Adjacent sequences:A006514A006515A006516*A006518A006519A006520

nonn,nice

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

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

approved