A139791 Numbers n for which 2n is a multiple of A002326(n), the multiplicative order of 2 mod 2n+1.

1, 2, 3, 5, 6, 8, 9, 11, 14, 15, 18, 20, 21, 23, 26, 29, 30, 33, 35, 36, 39, 41, 44, 48, 50, 51, 53, 54, 56, 63, 65, 68, 69, 74, 75, 78, 81, 83, 86, 89, 90, 95, 96, 98, 99, 105, 111, 113, 114, 116, 119, 120, 125, 128, 131, 134, 135, 138, 140, 141, 146, 153, 155, 156, 158, 165, 168, 170

Offset: 1

Author

Vladimir Shevelev, May 21 2008, May 24 2008

Keywords

• nonn

Comments

The sequence properly contains A005097. 170 is the first number which is not in A005097. One can prove that A002326(2^(2t-1)) = 4t. Thus if n=2^(2t-1), where, for any m>0, t=2^(m-1) then 2n is a multiple of A002326(n) while 2n+1 is a Fermat number which, as well known, is not always a prime.

The sequence is the union of A005097 and (A001567 - 1)/2. [Conjectured by Vladimir Shevelev, proved by Ray Chandler, May 26 2008]

References

• Christopher Adler and Jean-Paul Allouche (2022), Finite self-similar sequences, permutation cycles, and music composition, Journal of Mathematics and the Arts, 16:3, 244-261, DOI: 10.1080/17513472.2022.2116745.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A002326, A005097, A001262, A001567, A137576.

Programs

Mathematica

Select[Range[160], Divisible[2#, MultiplicativeOrder[2, 2#+1]] &] (* Amiram Eldar, Jun 28 2019 *)

PARI

isok(n) = !(2*n % znorder(Mod(2, 2*n+1))); \\ Michel Marcus, Nov 02 2017

Extensions

Data extended up to a(68) = 170 to clarify distinction from A005097 and essentially identical sequences A130290 and A102781, by M. F. Hasler, Dec 13 2019