A383131 a(n) is the number of iterations that n requires to reach 1 under the map x -> -x/2 if x is even, 3x + 1 if x is odd; a(n) = -1 if 1 is never reached.

0, 3, 12, 2, 5, 5, 8, 5, 11, 11, 50, 14, 14, 14, 53, 4, 17, 17, 43, 7, 7, 7, 20, 7, 46, 46, 59, 10, 10, 10, 23, 7, 49, 49, 62, 13, 13, 13, 13, 13, 26, 26, 39, 52, 52, 52, 65, 16, 16, 16, 78, 16, 16, 16, 29, 16, 42, 42, 55, 55, 55, 55, 68, 6, 19, 19, 19, 19, 19

Offset: 1

Author

Ya-Ping Lu, Apr 17 2025

Keywords

• nonn

Comments

A plot of a(n) for non-zero integers n in the range of (-10000, 10000) is given in LINKS.

Conjecture: a(n) != -1 for any positive or negative integer n.

Examples

			a(3) = 12 because it takes 12 steps for n = 3 to reach 1: 3 -> 10 -> -5 -> -14 -> 7 -> 22 -> -11 -> -32 -> 16 -> -8 -> 4 -> -2 -> 1.
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..10000
• Ya-Ping Lu, A plot of a(n) for n in the range of (-10000, 10000)

Crossrefs

Cf. A006577, A381055.

Programs

Maple

a:= proc(n) option remember; `if`(n=1, 0,
      1+a(`if`(n::even, -n/2, 3*n+1)))
    end:
seq(a(n), n=1..69);  # Alois P. Heinz, Aug 13 2025

Mathematica

js[n_] := If[EvenQ[n],-n/2,3n+1]; f[n_] := Length[ NestWhileList[js, n, # != 1 &]] - 1; Table[ f[n], {n, 69}] (* James C. McMahon, Apr 30 2025 *)

PARI

a(n) = { for (k = 0, oo, if (n==1, return (k), n = if (n%2, 3*n+1, -n/2));); } \\ Rémy Sigrist, Apr 19 2025

Python

def A383131(n):
    ct = 0
    while n != 1: n = 3*n+1 if n%2 else -n>>1; ct += 1
    return ct