A323460 Choix de Bruxelles, version 2: irregular table read by rows in which row n lists all the legal numbers that can be reached by halving or doubling some substring of the decimal expansion of n (including the empty string).

1, 2, 1, 2, 4, 3, 6, 2, 4, 8, 5, 10, 3, 6, 12, 7, 14, 4, 8, 16, 9, 18, 5, 10, 20, 11, 12, 21, 22, 6, 11, 12, 14, 22, 24, 13, 16, 23, 26, 7, 12, 14, 18, 24, 28, 15, 25, 30, 110, 8, 13, 16, 26, 32, 112, 17, 27, 34, 114, 9, 14, 18, 28, 36, 116, 19, 29, 38

Offset

1,2

Comments

The differs from the first version (in A323286) in that now n can be reached from n (by using the empty string).

This slight modification of the definition makes the analysis simpler.

The number of numbers that can be reached from n in one step is A323287(n)+1.

The minimal number of steps to reach n starting at 1 is still given by A323454.

Links

Table of n, a(n) for n=1..68.

Eric Angelini, Lars Blomberg, Charlie Neder, Remy Sigrist, and N. J. A. Sloane, "Choix de Bruxelles": A New Operation on Positive Integers, arXiv:1902.01444 [math.NT], Feb 2019; Fib. Quart. 57:3 (2019), 195-200.

Eric Angelini, Lars Blomberg, Charlie Neder, Remy Sigrist, and N. J. A. Sloane,, "Choix de Bruxelles": A New Operation on Positive Integers, Local copy.

Brady Haran and N. J. A. Sloane, The Brussels Choice, Numberphile video (2020)

Example

Rows 1 through 20 are:
1, 2,
1, 2, 4,
3, 6,
2, 4, 8,
5, 10,
3, 6, 12,
7, 14,
4, 8, 16,
9, 18,
5, 10, 20,
11, 12, 21, 22,
6, 11, 12, 14, 22, 24,
13, 16, 23, 26,
7, 12, 14, 18, 24, 28,
15, 25, 30, 110,
8, 13, 16, 26, 32, 112,
17, 27, 34, 114,
9, 14, 18, 28, 36, 116,
19, 29, 38, 118,
10, 20, 40

Prog

(Python)
def cdb2(n):
    s, out = str(n), {n}
    for l in range(1, len(s)+1):
        for i in range(len(s)+1-l):
            if s[i] == '0': continue
            t = int(s[i:i+l])
            out.add(int(s[:i] + str(2*t) + s[i+l:]))
            if t&1 == 0: out.add(int(s[:i] + str(t//2) + s[i+l:]))
    return sorted(out)
print([c for n in range(1, 21) for c in cdb2(n)]) # Michael S. Branicky, Jul 24 2022

Crossrefs

Cf. A323286, A323287, A323453, A323454.

Sequence in context: A364780 A124904 A187500 * A129144 A360594 A295313

Adjacent sequences: A323457 A323458 A323459 * A323461 A323462 A323463

Keyword

nonn,base,tabf

Author

N. J. A. Sloane, Jan 22 2019

Status

approved