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A054085 a(n) = order of n in the permutation A054084 of the natural numbers if this order exists; a(n) = -1 otherwise.

%I A054085 #17 Nov 28 2024 13:27:45
%S A054085 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 A054085 -1,-1,16,16,16,16,16,-1,-1,16,16,-1,16,-1,16,-1,-1,16,-1,-1,16,-1,16,
%U A054085 -1,-1,-1,-1,-1,-1,-1,-1,-1,16,-1,-1,-1,-1,-1,-1,-1
%N A054085 a(n) = order of n in the permutation A054084 of the natural numbers if this order exists; a(n) = -1 otherwise.
%e A054085 3 -> 5 -> 7 -> 10 -> 8 -> 6 -> 4 -> 3, so that 3 has order 7.
%t A054085 a054084[n_] := a054084[n] = If[EvenQ[n], Floor[n/2*GoldenRatio],
%t A054085 Floor[(n + 1)/2*GoldenRatio^2]];
%t A054085 Array[a054084[#] &, 40]  (* after _Jean-François Alcover_ *)
%t A054085 Table[Length[NestWhileList[a054084, a054084[n], # != n &, 1,
%t A054085      10000]] /. (10001 -> -1), {n, 1, 500}]
%t A054085 (* _Peter J. C. Moses_, Jan 26 2022 *)
%Y A054085 Cf. A054083, A054084.
%K A054085 sign
%O A054085 1,1
%A A054085 _Clark Kimberling_
%E A054085 Data truncated by _Sean A. Irvine_, Jan 23 2022
%E A054085 Edited by _Clark Kimberling_, Jan 26 2022