Search: id:A006521
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%I A006521 M2806
%S A006521 1,3,9,27,81,171,243,513,729,1539,2187,3249,4617,6561,9747,13203,13851,
%T A006521 19683,29241,39609,41553,59049,61731,87723,97641,118827,124659,177147,
%U A006521 185193,250857,263169,292923,354537,356481,373977,531441,555579,752571
%N A006521 Numbers n such that n divides 2^n + 1.
%C A006521 Closed under multiplication: if x and y are terms then so it x*y.
%C A006521 More is true: 1. If n is in the sequence then so is any multiple of n 
               having the same prime factors as n. 2. If n and m are in the sequence 
               then so is lcm(n,m). For a proof, see [1]. Elements of the sequence 
               that cannot be generated from smaller elements of the sequence using 
               either of these rules are called *primitive*. The sequence of primitive 
               solutions of n|2^n+1 is A136473. 3. The sequence satisfies various 
               congruences, which enable it to be generated quickly. For instance, 
               every element of this sequence not a power of 3 is divisible either 
               by 171 or 243 or 13203 or 2354697 or 10970073 or 22032887841. See 
               the Bailey-Smyth reference. - Toby Bailey and Chris Smyth (c.smyth(AT)ed.ac.uk), 
               Jan 13 2008
%D A006521 J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 243, p. 68, Ellipses, 
               Paris 2008.
%D A006521 R. Honsberger, Mathematical Gems, M.A.A., 1973, p. 142.
%D A006521 Sierpinski, W. 250 Problems in Elementary Number Theory. New York: American 
               Elsevier, 1970. Problem #16.
%D A006521 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%H A006521 Robert G. Wilson v, <a href="b006521.txt">Table of n, a(n) for n = 1..518 
               </a>. [From Robert G. Wilson v (rgwv(AT)rgwv.com), Jul 06 2009]
%H A006521 Toby Bailey and Chris Smyth <a href="http://www.maths.ed.ac.uk/~chris/
               n_divides_2to_nplus1.pdf">Primitive solutions of n|2^n+1</a>.
%p A006521 for n from 1 to 1000 do if 2^n +1 mod n = 0 then lprint(n); fi; od;
%p A006521 S:=1,3,9,27,81:C:={171,243,13203,2354697,10970073,22032887841}: for c 
               in C do for j from c to 10^8 by 2*c do if 2&^j+1 mod j = 0 then S:=S, 
               j;fi;od;od; S:=op(sort([op({S})])); - Toby Bailey and Chris Smyth 
               (c.smyth(AT)ed.ac.uk), Jan 13 2008
%t A006521 Do[If[PowerMod[2, n, n] + 1 == n, Print[n]], {n, 1, 10^6}]
%t A006521 k = 9; lst = {1, 3}; While[k < 1000000, a = PowerMod[2, k, k]; If[a + 
               1 == k, AppendTo[lst, k]]; k += 9]; lst [From Robert G. Wilson v 
               (rgwv(AT)rgwv.com), Jul 06 2009]
%Y A006521 Cf. A006517.
%Y A006521 Cf. A057719 (prime factors of numbers in A006521) A136473 (primitive 
               n such that n divides 2^n+1).
%Y A006521 Sequence in context: A036145 A014950 A036143 this_sequence A014953 A080557 
               A022014
%Y A006521 Adjacent sequences: A006518 A006519 A006520 this_sequence A006522 A006523 
               A006524
%K A006521 nonn
%O A006521 1,2
%A A006521 N. J. A. Sloane (njas(AT)research.att.com).
%E A006521 More terms from David W. Wilson (davidwwilson(AT)comcast.net)

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