This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A339769 #11 Feb 01 2021 13:46:45
%S A339769 1,2,4,4,5,6,10,12,9,7,4,9,13,11,7,6,8,10,13,14,10,15,14,21,8,7,13,21
%N A339769 Number of unique heights that are achieved by only one starting number in the Collatz (or '3x+1') problem when starting from numbers in the range [2^n, 2^(n+1)).
%C A339769 The number of unique heights in each interval [2^n, 2^(n+1)) appears to be linear in n - with large oscillations about the best fit line - while the values of attained maximum heights appear to be quadratic in n.
%C A339769 Neither this sequence nor the corresponding sequence of maximum heights in each interval is monotone, e.g.: a(23..27) = (21, 8, 7, 13, 21) and the corresponding maximum heights are (704, 702, 949, 947, 956).
%e A339769 a(5)=6 since the 6 heights 5, 26, 34, 109, 29, 104 are uniquely attained from the starting numbers 32, 33, 39, 41, 43, 47, respectively. The largest of the distinct heights (A280341) in the interval [32,64) however is 112.
%e A339769 a(11)=9 with largest unique height 237 for starting value 3711 in interval [2^11, 2^12) also is the largest height for all starting values in the interval.
%t A339769 collatz[n_] := If[EvenQ[n], n/2, 3n+1]
%t A339769 height[n_] := Length[NestWhileList[collatz, n, #!=1&]] - 1
%t A339769 a339769[n_] := Module[{heightL={}, countL={}, s, h, p}, For[s=2^n, s<2^(n+1), s++, h=height[s]; If[!MemberQ[heightL, h], AppendTo[heightL, h]; AppendTo[countL, 1], {{p}}=Position[heightL, h]; countL[[p]]+=1]]; Length[Select[Transpose[{heightL, countL}], #[[2]]==1&]]]
%t A339769 (* sequence data; long computation times for n >= 22 *)
%t A339769 Map[a339769, Range[0, 27]]
%Y A339769 Cf. A006577, A277109, A280341.
%K A339769 nonn,more
%O A339769 0,2
%A A339769 _Hartmut F. W. Hoft_, Dec 16 2020