A390816 Number of tripling steps in the Collatz 3x+1 iteration going from 2^n-1 to 1.

0, 2, 5, 5, 39, 39, 15, 15, 20, 20, 56, 56, 56, 56, 44, 44, 80, 80, 61, 61, 109, 109, 174, 174, 162, 162, 138, 138, 162, 162, 162, 162, 191, 191, 191, 191, 191, 191, 191, 191, 191, 191, 210, 210, 210, 210, 210, 210, 210, 210, 210, 210, 309, 309, 210, 210, 309

Offset

1,2

Links

Table of n, a(n) for n=1..57.

Formula

a(n) = A006667(A000225(n)).

a(n) = A075680(2^(n-1)).

Example

For n=4, 2^4-1 = 15 has a(4)=5 tripling steps on its way to 1.

Maple

b:= proc(n) option remember; `if`(n<2, 0,
      `if`(n::even, b(n/2), 1+b(3*n+1)))
    end:
a:= n-> b(2^n-1):
seq(a(n), n=1..57);  # Alois P. Heinz, Nov 21 2025

Mathematica

a[n_] := -1 + Count[NestWhileList[If[OddQ[#], 3*# + 1, #/2] &, 2^n - 1, # > 1 &], _?OddQ]; Array[a, 55] (* Amiram Eldar, Nov 21 2025 *)

Prog

(Python)
def a(n: int) -> int:
    x = (1 << n) - 1
    steps = 0
    while x != 1:
        if x % 2 == 0:
            x //= 2
        else:
            x = 3*x + 1
            steps += 1
    return steps

Crossrefs

Cf. A000225, A006667, A075680, A193688, A181777, A390817.

Sequence in context: A154918 A176862 A352392 * A176081 A271222 A073339

Adjacent sequences: A390813 A390814 A390815 * A390817 A390818 A390819

Keyword

nonn

Author

Stephen R. Campbell, Nov 20 2025

Status

approved