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

1,3

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

Formula

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

Prog

(PARI) See Links section.

Crossrefs

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

Sequence in context: A074457 A200237 A072761 * A340420 A127885 A006577

Adjacent sequences: A337354 A337355 A337356 * A337358 A337359 A337360

Keyword

nonn,base

Author

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

Status

approved