A337357 "Choix de Collatz": a(n) is the least number of steps required to reach 1 starting from n under substring substitutions of the form k -> T(k) (where T is the Collatz map, A006370).

0, 1, 7, 2, 5, 8, 9, 3, 11, 6, 7, 8, 9, 9, 7, 4, 7, 10, 8, 7, 7, 8, 10, 9, 8, 5, 7, 10, 8, 8, 7, 5, 11, 6, 9, 11, 6, 7, 8, 8, 7, 8, 11, 9, 9, 6, 8, 10, 8, 9, 8, 6, 11, 7, 9, 9, 9, 8, 11, 9, 7, 6, 11, 6, 9, 12, 7, 7, 9, 10, 8, 7, 10, 7, 10, 8, 10, 8, 10, 9, 8

Offset: 1

Author

Rémy Sigrist and N. J. A. Sloane, Aug 24 2020

Keywords

• nonn
• base

Comments

This sequence is a variant of "Choix de Bruxelles" (where we consider substring substitutions of the form k <-> 2*k, see A323286):

- we map a positive number n to any number that can be obtained as follows:

- take a nonempty substring s (without leading zero) in the decimal representation of n,

- if the value of s corresponds to an even number, replace s by s/2,

- otherwise replace s by 3*s + 1.

The sequence is well defined:

- the proof is similar to that described in A337321,

- the initial paths to consider here are the following:

1

2 -> 1

3 -> 10 -> 5 -> 16 -> 8 -> 4 -> 2 -> 1

4 -> 2 -> 1

5 -> 16 -> 8 -> 4 -> 2 -> 1

6 -> 3 -> 10 -> 5 -> 16 -> 8 -> 4 -> 2 -> 1

7 -> 22 -> 11 -> 34 -> 32 -> 16 -> 8 -> 4 -> 2 -> 1

8 -> 4 -> 2 -> 1

9 -> 28 -> 24 -> 22 -> 21 -> 64 -> 32 -> 16 -> 8 -> 4 -> 2 -> 1

10 -> 5 -> 16 -> 8 -> 4 -> 2 -> 1

11 -> 34 -> 32 -> 16 -> 8 -> 4 -> 2 -> 1

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..3652
• Rémy Sigrist, PARI program for A337357

Crossrefs

Cf. A006370, A006577, A323286, A323454, A337321.

Programs

PARI

See Links section.

Formula

a(n) <= A006577(n) (when A006577(n) >= 0).