A138004 Odd numbers k for which all nontrivial cycles of the Ducci map on k-tuples modulo 2 have the same length.

3, 5, 7, 11, 13, 19, 23, 29, 31, 37, 43, 47, 53, 59, 61, 67, 71, 79, 83, 89, 101, 103, 107, 127, 131, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 197, 199, 211, 223, 227, 233, 239, 251, 263, 269, 271, 281, 283, 293, 307, 311, 313, 317, 347, 349, 359, 367, 373, 379, 383, 389

Offset: 1

Author

T. D. Noe, Feb 26 2008

Keywords

• nonn

Comments

All nontrivial cycles have the same length when either k is a prime number with primitive root 2 (see A001122), or when all factors of polynomial ((x+1)^k+1)/x (mod 2) have the same multiplicative order. The corresponding cycle lengths are listed in A138006.

It is conjectured that all terms of this sequence are prime numbers.

Numbers k such that A111944(k) = 2. - Max Alekseyev, Jul 10 2025

Links

• Max Alekseyev, Table of n, a(n) for n = 1..310
• Florian Breuer, Ernest Lotter and Brink van der Merve, Ducci sequences and cyclotomic polynomials, Finite Fields Appl. 13 (2007), 293-304.
• Michal Misiurewicz, John G. Stevens and Diana M. Thomas, Iterations of Linear Maps over Finite Fields, Linear Algebra and its Applications, Vol. 413 (2006), 218-234.

Crossrefs

Cf. A001122, A038553, A111944, A138005, A138006.

Extensions

Edited and terms a(36) onward added by Max Alekseyev, Jul 10 2025