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%I #17 Mar 26 2019 19:43:00
%S 1,2,4,7,21,49,85,253,442,766,1327,2299,3982,11839,20506,35518,61519,
%T 184557,553645,966928,1674769,2900785,8701141,25877593,44821306,
%U 77676682,134539960,402368674,696922987,1207106023,2090768962,3632578906
%N Numbers with record number of iterations of x -> A306938(x) required to reach 1 (A306944).
%C The record numbers of iterations are 0, 2, 4, 6, 7, 8, 10, 11, 13, 15, 17, 19, 21, 22, 24, 26, 28, 29, 30, 32, 34, 36, 37, 38, 40, 42, 44, 45, 47, 49, 51, 53.
%C Conjecture: sqrt(3) < a(n)/a(n-1) <= 3. - _Daniel Suteu_, Mar 20 2019
%H H. J. J. te Riele, <a href="https://ir.cwi.nl/pub/6662">Iteration of number-theoretic functions</a>, Nieuw Archief v. Wiskunde, (4) 1 (1983), 345-360. See Example III.3.
%e 2 reaches 1 after 2 iterations: 2 -> 3 -> 1;
%e 4 reaches 1 after a record number of 4 iterations: 4 -> 6 -> 2 -> 3 -> 1.
%t f[n_] := If[Divisible[n, 3], n/3, Floor[n*Sqrt[3]]]; itnum[n_] := Module[{k = n, inc = False, c = 0}, While[k > 1, kk = f[k]; If[inc && kk > k, c = -1; Break[]]; inc = kk > k; k = kk; c++]; c]; s={}; im=-1; Do[i=itnum[n]; If[i>im, im=i; AppendTo[s, n]], {n,1,10000}]; s
%Y Cf. A306938, A306943, A306944.
%K nonn,more
%O 1,2
%A _Amiram Eldar_, Mar 18 2019