A336914 Number of steps to reach 1 in '3^x+1' problem (a variation of the Collatz problem), or -1 if 1 is never reached.

0, 1, 4, 2, 11, 2, 9, 5, 7, 5, 7, 5, 5, 5, 16, 3, 5, 3, 5, 3, 16, 3, 14, 3, 9, 3, 14, 3, 9, 3, 9, 12, 14, 12, 22, 12, 14, 12, 7, 12, 5, 12, 5, 12, 7, 12, 5, 12, 7, 12, 5, 12, 5, 12, 20, 12, 5, 12, 16, 12, 5, 12, 14, 3, 12, 3, 5, 3, 14, 3, 5, 3, 14, 3, 5, 3, 5

Offset

1,3

Comments

The 3^x+1 map, which is a variation of the 3x+1 (Collatz) map, is defined for x >= 1 as follows: if x is odd, then map x to 3^x+1; otherwise, map x to floor(log_2(x)).

It seems that all 3^x+1 trajectories reach 1; this has been verified up to 10^9.

Links

Table of n, a(n) for n=1..77.

Wikipedia, Collatz conjecture

Example

For n = 5, a(5) = 11, because there are 11 steps from 5 to 1 in the following trajectory for 5: 5, 244, 7, 2188, 11, 177148, 17, 129140164, 26, 4, 2, 1.
For n = 6, a(6) = 2, because there are 2 steps from 6 to 1 in the following trajectory for 6: 6, 2, 1.

Prog

(Python)
from math import floor, log
def a(n):
    if n == 1: return 0
    count = 0
    while True:
        if n % 2: n = 3**n + 1
        else: n = int(floor(log(n, 2)))
        count += 1
        if n == 1: break
    return count
print([a(n) for n in range(1, 101)])

Crossrefs

Cf. A006370 (image of n under the 3x+1 map).

Cf. A336913 (image of n under the 3^x+1 map).

Sequence in context: A135440 A215500 A188128 * A091484 A163544 A191728

Adjacent sequences: A336911 A336912 A336913 * A336915 A336916 A336917

Keyword

nonn

Author

Robert C. Lyons, Aug 08 2020

Status

approved