A393248 a(n) is the period length of the Ducci sequence, where the initial vector consists of the binary digits of n.

1, 1, 1, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 1, 6, 6, 6, 3, 3, 6, 6, 6, 6, 6, 1, 6, 6, 3, 6, 6, 6, 3, 6, 6, 6, 6, 3, 6, 3, 6, 6, 6, 6, 6, 6, 1, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7

Offset

1,4

Links

Table of n, a(n) for n=1..80.

Wikipedia, Ducci sequence.

Formula

Let A_k = {a(n) | 2^(k-1) <= n < 2^k}. Then max(A_k) = A038553(k) and #A_k = A111944(k). - Andrei Zabolotskii, Feb 14 2026

Example

n = 6: 6_10 = 110_2, vectors are (1,1,0)-->(0,1,1)-->(1,0,1)-->(1,1,0) repeats after 3 steps, thus a(6) = 3.
n = 7: 7_10 = 111_2, vectors are (1,1,1)-->(0,0,0)-->(0,0,0) repeats after 1 step, thus a(7) = 1.

Mathematica

a[n_] := Length[#] - FirstPosition[#, Last[#]][[1]]& @ NestWhileList[Abs[# - RotateLeft[#]] &, IntegerDigits[n, 2], Unequal, All]; Array[a, 80] (* Amiram Eldar, Feb 07 2026 *)

Crossrefs

Cf. A007788, A038553, A111944.

Sequence in context: A380219 A066601 A110566 * A126066 A177693 A353631

Adjacent sequences: A393245 A393246 A393247 * A393249 A393250 A393251

Keyword

nonn,base

Author

Ctibor O. Zizka, Feb 07 2026

Status

approved