A280341 Number of distinct heights achieved in the Collatz (or '3x+1') problem when starting from numbers in the range [2^n,2^(n+1)).

1, 2, 4, 6, 10, 16, 26, 41, 53, 64, 74, 85, 101, 118, 128, 144, 157, 174, 195, 217, 238, 261, 281, 309, 324, 342, 364, 397

Offset: 0

Author

Dmitry Kamenetsky, Jan 01 2017

Keywords

• nonn

Comments

Here the height is defined to be the number of halving and tripling steps required to reach 1.

Interestingly the values in this sequence grow slowly (almost linearly) indicating that the average number of starting values with the same height increases with n.

Question: Is this sequence always increasing?

Definition corrected by N. J. A. Sloane, Apr 09 2020. The old definition was "Number of unique heights achieved in the Collatz (or '3x+1') problem when starting from numbers in the range [2^n,2^(n+1))."

This sequence a(n) as well as the sequence of maximum heights in each interval appear to increase quadratically with n. The odd numbers in [2^n, 2^(n+1)) , 5 <= n <= 20, create all distinct heights for the interval except for height n of number 2^n, and except for height n+3 when n is odd. - Hartmut F. W. Hoft, Dec 16 2020

Examples

			The heights for starting values 16 to 31 are: 4, 12, 20, 20, 7, 7, 15, 15, 10, 23, 10, 111, 18, 18, 18, 106. The unique heights are: 4, 12, 20, 7, 15, 10, 23, 111, 18, 106. Hence a(4)=10.
		

Crossrefs

Cf. A006577, A277109.

Programs

Mathematica

collatz[n_] := If[EvenQ[n], n/2, 3n+1]
height[n_] := Length[NestWhileList[collatz, n, #!=1&]] - 1
a280341[n_] := Length[Union[Map[height, Range[2^n, 2^(n+1)-1]]]]
(* sequence data; long computation times for n >= 22 *)
Map[a280341, Range[0, 27]]
(* Hartmut F. W. Hoft, Dec 16 2020 *)

Extensions

a(25)-a(27) from Hartmut F. W. Hoft, Dec 16 2020