1,2

Closed under multiplication: if x and y are terms then so it x*y.

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

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 243, p. 68, Ellipses, Paris 2008.

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

Sierpinski, W. 250 Problems in Elementary Number Theory. New York: American Elsevier, 1970. Problem #16.

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

Robert G. Wilson v, Table of n, a(n) for n = 1..518 . [From Robert G. Wilson v (rgwv(AT)rgwv.com), Jul 06 2009]

Toby Bailey and Chris Smyth Primitive solutions of n|2^n+1.

for n from 1 to 1000 do if 2^n +1 mod n = 0 then lprint(n); fi; od;

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

Do[If[PowerMod[2, n, n] + 1 == n, Print[n]], {n, 1, 10^6}]

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]

Cf. A006517.

Cf. A057719 (prime factors of numbers in A006521) A136473 (primitive n such that n divides 2^n+1).

Sequence in context: A036145 A014950 A036143 this_sequence A014953 A080557 A022014

Adjacent sequences: A006518 A006519 A006520 this_sequence A006522 A006523 A006524

nonn

N. J. A. Sloane (njas(AT)research.att.com).

More terms from David W. Wilson (davidwwilson(AT)comcast.net)