A224254 Full cycle lengths in the Collatz (3x+1) problem when the negative integers are used.

2, 2, 2, 2, 5, 2, 5, 2, 5, 5, 2, 2, 5, 5, 2, 2, 18, 5, 5, 5, 18, 2, 18, 2, 18, 5, 5, 5, 2, 2, 18, 2, 18, 18, 5, 5, 18, 5, 2, 5, 18, 18, 2, 2, 18, 18, 5, 2, 18, 18, 5, 5, 2, 5, 18, 5, 2, 2, 2, 2, 18, 18, 5, 2, 2, 18, 18, 18, 2, 5, 2, 5, 18, 18, 5, 5, 2, 2, 2, 5, 5

Offset

1,1

Comments

There are other cycles of lengths 2, 5 and 18 if negative integers are used. In Z, it is conjectured that the five values of cycle are 1, 2, 3, 5 and 18 (see A121510).

Links

Table of n, a(n) for n=1..81.

Wikipedia, Collatz conjecture

Example

a(1) = 2 because the cycle -1 -> -2 -> -1... contains 2 distinct terms;
a(5) = 5 because the cycle -5 -> -14 -> -7->-20 -> -5 ... contains 5 distinct terms;
a(17) = 18 because the cycle -17 -> -50 -> -25->-74 -> -37 -> -110 -> -55->-164 -> -82 -> -41 -> -122->-61 -> -182 -> -91 -> -272->-136 -> -68 -> -34 -> -17... contains 18 distinct terms.

Prog

(Python)
import sympy
def A224254(n):
  return next(sympy.cycle_length(lambda x:3*x+1 if x%2 else x//2, -n))[0] # Pontus von Brömssen, Jan 24 2021

Crossrefs

Cf. A121510, A224166, A224183.

Sequence in context: A362034 A154097 A221491 * A107604 A080647 A324516

Adjacent sequences: A224251 A224252 A224253 * A224255 A224256 A224257

Keyword

nonn

Author

Michel Lagneau, Apr 02 2013

Extensions

Data corrected by Pontus von Brömssen, Jan 24 2021

Status

approved