A054085 -id:A054085 - cp's OEIS frontend

A054084 Permutation of N: for each k >= 1, let p(k)=least natural number not already an a(i), q(k)=p(k)+k, a(2k-1)=q(k), a(2k)=p(k).

2, 1, 5, 3, 7, 4, 10, 6, 13, 8, 15, 9, 18, 11, 20, 12, 23, 14, 26, 16, 28, 17, 31, 19, 34, 21, 36, 22, 39, 24, 41, 25, 44, 27, 47, 29, 49, 30, 52, 32, 54, 33, 57, 35, 60, 37, 62, 38, 65, 40, 68, 42, 70, 43, 73, 45, 75, 46, 78, 48, 81, 50, 83

Offset: 1

Clark Kimberling

Index entries for sequences that are permutations of the natural numbers

Cf. A054082, A054085.

Odd-indexed terms: A001950 (Upper Wythoff sequence). Even-indexed terms: A000201 (Lower Wythoff sequence). Inverse permutation: A064786.

from math import isqrt
def A054084(n): return ((m:=n+1>>1)+isqrt(5*m**2)>>1)+m*(n&1) # Chai Wah Wu, Aug 25 2022

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

Original entry on oeis.org

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

Clark Kimberling

sign

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)

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

Cf. A054082, A054085.

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 *)

Data truncated by Sean A. Irvine, Jan 23 2022

Edited by Clark Kimberling, Jan 26 2022

cp's OEIS Frontend

A054084 Permutation of N: for each k >= 1, let p(k)=least natural number not already an a(i), q(k)=p(k)+k, a(2k-1)=q(k), a(2k)=p(k).

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

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