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A359214 a(n) is the least k >= 0 such that A359194^k(A358668(n)) = n (where A359194^k denotes the k-th iterate of A359194).

%I A359214 #33 Dec 23 2022 11:22:34
%S A359214 0,1,0,0,0,0,1,0,0,0,1,0,0,2,0,4,3,0,5,0,0,1,74,0,3,7,0,1,0,0,1,5,0,0,
%T A359214 6,0,0,2,0,77,1,0,0,1,0,0,2,0,0,1,0,0,8,0,0,4,0,9,1,0,0,75,0,7,6,0,8,
%U A359214 0,0,1,0,0,76,0,0,1,5418,0,1,0,0,2,0,0
%N A359214 a(n) is the least k >= 0 such that A359194^k(A358668(n)) = n (where A359194^k denotes the k-th iterate of A359194).
%H A359214 Rémy Sigrist, <a href="/A359214/b359214.txt">Table of n, a(n) for n = 0..10000</a>
%H A359214 Rémy Sigrist, <a href="/A359214/a359214.gp.txt">PARI program</a>
%F A359214 a(n) = 0 iff A358668(n) = n.
%F A359214 a(3*n+2) = 0. - _Thomas Scheuerle_, Dec 22 2022
%e A359214 The orbit of 0 under repeated application of A359194 is:
%e A359214     0, 1, 0, ...
%e A359214 So a(0) = 0, a(1) = 1.
%e A359214 The orbit of 2 under repeated application of A359194 is:
%e A359214     2, 1, 0, 1, 0, ...
%e A359214 So a(2) = 0.
%e A359214 The orbit of 3 under repeated application of A359194 is:
%e A359214     3, 6, 13, 24, 55, 90, 241, 300, 123, 142, 85, 0, 1, 0, ...
%e A359214 So a(3) = 0, a(6) = 1, a(13) = 2, a(24) = 3, a(55) = 4, etc.
%t A359214 nn = 83; c[_] = -1; c[0] = 0; f[n_] := FromDigits[BitXor[1, IntegerDigits[3*n, 2]], 2]; Do[(MapIndexed[If[c[#1] == -1, Set[c[#1], First[#2] - 1]] &, #]; -1 + Length[#]) &@ NestWhileList[f, n, c[#] == -1 && # > 1 &], {n, 0, nn}]; Array[c, nn] (* _Michael De Vlieger_, Dec 23 2022 *)
%o A359214 (PARI) See Links section.
%Y A359214 Cf. A358668, A359194, A359226.
%Y A359214 Cf. A343858 (smallest numbers inside cyclic trajectories of the generalized Collatz function bx+c).
%K A359214 nonn,base
%O A359214 0,14
%A A359214 _Rémy Sigrist_, Dec 22 2022