A054083 a(n) = order of n in the permutation A054082 of the natural numbers if this order exists; a(n) = -1 otherwise.

2, 2, 2, 2, 7, 7, 7, 7, 7, 7, 9, 7, 9, 9, 9, 9, 9, 9, -1, 9, -1, 9, -1, -1, -1, -1, 16, -1, 16, -1, 16, -1, -1, 16, 16, 16, 16, 16, -1, -1, 16, 16, -1, 16, -1, 16, -1, -1, 16, -1, -1, 16, -1, 16, -1, -1, -1, -1, -1, -1, -1, -1, -1, 16, -1, -1, -1, -1, -1, -1

Offset: 1

Author

Clark Kimberling

Keywords

• sign

Comments

From Peter J. C. Moses, Jan 26 2022: (Start)

For n up to 60000,

a(n) = 2 for n = 1, 2, 3, 4;

a(n) = 7 for n = 5, 6, 7, 8, 9, 10, 12

a(n) = 9 for n = 11, 13, 14, 15, 16, 17, 18, 20, 22

a(n) = 16 for n = 27, 29, 31, 34, 35, 36, 37, 38, 41, 42, 44, 46, 49, 52, 54, 64

a(n) = 25 for n = 267, 283, 330, 343, 350, 371, 385, 393, 408, 424, 449, 467, 476, 486, 495, 504, 515, 524, 545, 578, 588, 612, 648, 674, 714. (End)

Examples

			5 -> 7 -> 9 -> 12 -> 10 -> 8-> 6-> 5, so that 5 has order 7.
		

Crossrefs

Cf. A054082, A054085.

Programs

Mathematica

a054082[n_] := a054082[n] = If[OddQ[n], Floor[((n + 1)/2 - 1) GoldenRatio] + (n + 1)/2 + 1,
Floor[(n/2 - 1) GoldenRatio] + 2]; a054082[2] = 1;
Array[a054082[#] &, 40]  (* after Jean-François Alcover *)
Table[Length[NestWhileList[a054082, a054082[n], # != n &, 1,
    10000]] /. (10001 -> -1), {n, 1, 500}]
(* Peter J. C. Moses, Jan 26 2022 *)

Extensions

Data truncated by Sean A. Irvine, Jan 23 2022

Edited by Clark Kimberling, Jan 26 2022