A384327 Minimal Trips Around The Collatz Galaxy: a(n) is the minimal cycle length containing n. Each step in the cycle must be either to the next larger integer or follow a Collatz trajectory: k -> 3k+1 if k is odd or k -> k/2 if k is even.

2, 2, 3, 3, 4, 4, 5, 4, 5, 4, 5, 5, 6, 5, 6, 4, 6, 6, 7, 6, 7, 5, 6, 6, 8, 7, 8, 5, 7, 7, 8, 8, 9, 6, 7, 7, 9, 9, 10, 6, 8, 8, 9, 9, 11, 7, 8, 8, 10, 10, 11, 7, 9, 9, 10, 10, 12, 8, 9, 9, 11, 11, 12, 8, 10, 10, 11, 11, 13, 9, 10, 10, 12, 12, 13, 9, 11, 11, 12

Offset: 1

Author

Gordon Hamilton, May 26 2025

Keywords

• nonn

Comments

Excellent puzzle to get elementary students engaged in arithmetic. Recommended for use after the Collatz conjecture (3x+1 conjecture) is introduced side-by-side with its 3x-1 cousin.

In a tree of possible choices from n every positive integer occurs at most once which allows for lots of cutting in the search space. - David A. Corneth, May 26 2025

Examples

			a(25) is 8 because the shortest cycle containing 25 has 8 elements: 25 => 26 => 13 => 14 => 15 => 46 => 23 => 24 => 25.
		

Links

• David A. Corneth, Table of n, a(n) for n = 1..10000
• David A. Corneth, list of n, a(n) and the (or a) shortest path of length a(n)
• David A. Corneth, PARI program
• Index entries for sequences related to 3x+1 (or Collatz) problem

Crossrefs

Cf. A006370.

Programs

PARI

\\ See Corneth link

Extensions

More terms from Alois P. Heinz, May 26 2025