%I #10 Jan 26 2022 11:09:35
%S 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,
%T -1,-1,16,16,16,16,16,-1,-1,16,16,-1,16,-1,16,-1,-1,16,-1,-1,16,-1,16,
%U -1,-1,-1,-1,-1,-1,-1,-1,-1,16,-1,-1,-1,-1,-1,-1,-1
%N a(n) = order of in the permutation A054084 of the natural numbers if this order exists; a(n) = -1 otherwise.
%e 3 -> 5 -> 7 -> 10 -> 8 -> 6 -> 4 -> 3, so that 3 has order 7.
%t a054084[n_] := a054084[n] = If[EvenQ[n], Floor[n/2*GoldenRatio],
%t Floor[(n + 1)/2*GoldenRatio^2]];
%t Array[a054084[#] &, 40] (* after Jean-François Alcover *)
%t Table[Length[NestWhileList[a054084, a054084[n], # != n &, 1,
%t 10000]] /. (10001 -> -1), {n, 1, 500}]
%t (* _Peter J. C. Moses_, Jan 26 2022 *)
%Y Cf. A054083, A054084.
%K sign
%O 1,1
%A _Clark Kimberling_
%E Data truncated by _Sean A. Irvine_, Jan 23 2022
%E Edited by _Clark Kimberling_, Jan 26 2022