A355187 Number of Collatz trajectories (A070165) for all positive integers <= 10^n that contain 2^4 as the greatest power of 2 within its trajectory.

6, 89, 933, 9401, 93744, 937712, 9379078, 93773848

Offset: 1

Author

Frank M Jackson, Jun 23 2022

Keywords

• nonn
• more

Comments

It is conjectured that lim_{n->infinity} a(n)/10^n = 15/16. Empirically, 93.75% of all trajectories have 2^4 as the greatest power of 2 within its trajectory. Sequence A135282(n) is the maximum power of 2 reached in the Collatz trajectory for integer n.

Examples

			a(1)=6 because the first 10 positive integers have trajectories, of which 6 have 2^4 as the greatest power of 2 in their trajectory.
These integers are 3, 5, 6, 7, 9, 10. See trajectory tables below.
  1:    1
  2:    2  1
  3:    3 10  5 16  8   4  2   1
  4:    4  2  1
  5:    5 16  8  4  2   1
  6:    6  3 10  5 16   8  4   2  1
  7:    7 22 11 34 17  52 26  13 40 20 10  5 16  8  4  2  1
  8:    8  4  2  1
  9:    9 28 14  7 22  11 34  17 52 26 13 40 20 10  5 16  8  4  2  1
  10:  10  5 16  8  4   2  1
		

Links

• Index entries for sequences related to 3x+1 (or Collatz) problem

Crossrefs

Cf. A070165, A135282.

Programs

Mathematica

collatz[n_] := Module[{}, If[OddQ[n], 3n+1, n/2]]; step[n_] := Module[{p=0, m=n, q}, While[!IntegerQ[q=Log[2, m]], m=collatz[m]; p++]; {p, q}]; Counts[Table[Last@step[n], {n, 1, 10^5}]][[Key[4]]]