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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.
1
6, 89, 933, 9401, 93744, 937712, 9379078, 93773848
OFFSET
1,1
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.
EXAMPLE
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
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]]]
CROSSREFS
Sequence in context: A127183 A054952 A101148 * A100297 A177568 A177573
KEYWORD
nonn,more
AUTHOR
Frank M Jackson, Jun 23 2022
STATUS
approved