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A006517 Numbers k such that k divides 2^k + 2.
(Formerly M1719)
21

%I M1719 #56 Feb 27 2024 01:09:56

%S 1,2,6,66,946,8646,180246,199606,265826,383846,1234806,3757426,

%T 9880278,14304466,23612226,27052806,43091686,63265474,66154726,

%U 69410706,81517766,106047766,129773526,130520566,149497986,184416166,279383126

%N Numbers k such that k divides 2^k + 2.

%C 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

%C 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

%C All terms greater than 6 have at least three distinct prime factors. - _Robert Israel_, Aug 21 2014

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

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

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

%H Giovanni Resta, <a href="/A006517/b006517.txt">Table of n, a(n) for n = 1..150</a>

%H Kin Y. Li et al., <a href="http://www.math.ust.hk/excalibur/v14_n2.pdf">Solution to Problem 323</a>, Mathematical Excalibur 14(2), 2009, p. 3.

%H V. Meally, <a href="/A006516/a006516.pdf">Letter to N. J. A. Sloane, May 1975</a>

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

%t Join[{1},Select[Range[28*10^7],PowerMod[2,#,#]==#-2&]] (* _Harvey P. Dale_, Aug 13 2018 *)

%o (PARI) is_A006517(n)=!(Mod(2,n)^n+2) \\ _M. F. Hasler_, Oct 08 2012

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

%K nonn,nice

%O 1,2

%A _N. J. A. Sloane_, _David W. Wilson_

%E Corrected and extended by Joe K. Crump (joecr(AT)carolina.rr.com), Sep 12 2000 and _Robert G. Wilson v_, Sep 13 2000

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Last modified March 29 03:51 EDT 2024. Contains 371264 sequences. (Running on oeis4.)