A336912 Number of steps to reach 1 in the 'x^3+1' problem (a variation of the Collatz problem), or -1 if 1 is never reached.

0, 1, 9, 2, 7, 2, 5, 2, 10, 10, 5, 10, 5, 10, 8, 3, 5, 3, 13, 3, 13, 3, 13, 3, 16, 8, 8, 8, 13, 8, 8, 8, 11, 8, 11, 3, 26, 3, 21, 3, 6, 3, 8, 3, 8, 3, 8, 3, 16, 6, 16, 6, 16, 6, 16, 6, 6, 6, 16, 6, 16, 6, 6, 3, 6, 3, 16, 3, 21, 3, 6, 3, 11, 3, 11, 3, 29, 3

Offset

1,3

Comments

The x^3+1 map, which is a variation of the 3x+1 (Collatz) map, is defined for x >= 0 as follows: if x is odd, then map x to x^3+1; otherwise, map x to floor(sqrt(x)).

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

Links

Table of n, a(n) for n=1..78.

Wikipedia, Collatz conjecture

Example

For n = 3, a(3) = 9, because there are 9 steps from 3 to 1 in the following trajectory for 3: 3, 28, 5, 126, 11, 1332, 36, 6, 2, 1.
For n = 4, a(4) = 2, because there are 2 steps from 4 to 1 in the following trajectory for 4: 4, 2, 1.

Prog

(Python)
from math import floor, sqrt
def a(n):
    if n == 1: return 0
    count = 0
    while True:
        if (n % 2) == 0: n = int(floor(sqrt(n)))
        else: n = n**3 + 1
        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. A336911 (image of n under the x^3+1 map).

Sequence in context: A224268 A019877 A252898 * A010537 A234371 A172423

Adjacent sequences: A336909 A336910 A336911 * A336913 A336914 A336915

Keyword

nonn

Author

Robert C. Lyons, Aug 07 2020

Status

approved