%I #8 Mar 20 2019 06:33:17

%S 7,23,63,175,459,1349,3506,8820,21897

%N The number of initial values <= 10^n whose trajectory under the iteration x -> A306938(x) reaches 1.

%C te Riele noted that of the first initial values <= 10^5 only 459 have a trajectory that reaches 1 and conjectured that almost all trajectories tend to infinity.

%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 a(1) = 7 since 7 of the initial values 1 to 10 (all except 5, 8, and 10) have a trajectory that reaches 1 (their A306944 value is not -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]; p=10; s={}; c=0; Do[i=itnum[n]; If[i>-1, c++]; If[n==p, AppendTo[s, c]; p*=10], {n,1,10000}]; s

%Y Cf. A306938, A306943, A306944.

%K nonn,more

%O 1,1

%A _Amiram Eldar_, Mar 18 2019