OFFSET
0,2
COMMENTS
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.
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).
EXAMPLE
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.
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.
MATHEMATICA
collatz[n_] := If[EvenQ[n], n/2, 3n+1]
height[n_] := Length[NestWhileList[collatz, n, #!=1&]] - 1
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&]]]
(* sequence data; long computation times for n >= 22 *)
Map[a339769, Range[0, 27]]
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Hartmut F. W. Hoft, Dec 16 2020
STATUS
approved