This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A006517 M1719 #56 Feb 27 2024 01:09:56
%S A006517 1,2,6,66,946,8646,180246,199606,265826,383846,1234806,3757426,
%T A006517 9880278,14304466,23612226,27052806,43091686,63265474,66154726,
%U A006517 69410706,81517766,106047766,129773526,130520566,149497986,184416166,279383126
%N A006517 Numbers k such that k divides 2^k + 2.
%C A006517 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 A006517 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 A006517 All terms greater than 6 have at least three distinct prime factors. - _Robert Israel_, Aug 21 2014
%D A006517 R. Honsberger, Mathematical Gems, M.A.A., 1973, p. 142.
%D A006517 W. Sierpiński, 250 Problems in Elementary Number Theory. New York: American Elsevier, 1970. Problem #18
%D A006517 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A006517 Giovanni Resta, <a href="/A006517/b006517.txt">Table of n, a(n) for n = 1..150</a>
%H A006517 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 A006517 V. Meally, <a href="/A006516/a006516.pdf">Letter to N. J. A. Sloane, May 1975</a>
%t A006517 Do[ If[ PowerMod[ 2, n, n ] + 2 == n, Print[n]], {n, 2, 1500000000, 4} ]
%t A006517 Join[{1},Select[Range[28*10^7],PowerMod[2,#,#]==#-2&]] (* _Harvey P. Dale_, Aug 13 2018 *)
%o A006517 (PARI) is_A006517(n)=!(Mod(2,n)^n+2) \\ _M. F. Hasler_, Oct 08 2012
%Y A006517 Cf. A006521, A015888, A015889, A015891, A015892, A015893, A015897, A015898, A015902, A015903, A015904, A015905, A015906.
%K A006517 nonn,nice
%O A006517 1,2
%A A006517 _N. J. A. Sloane_, _David W. Wilson_
%E A006517 Corrected and extended by Joe K. Crump (joecr(AT)carolina.rr.com), Sep 12 2000 and _Robert G. Wilson v_, Sep 13 2000