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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).
3
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
FORMULA
a(n) <= A006577(n) (when A006577(n) >= 0).
PROG
(PARI) See Links section.
CROSSREFS
KEYWORD
nonn,base
AUTHOR
STATUS
approved