A307635 -id:A307635 - cp's OEIS frontend

A323464 Values of n at which A323484 reaches a new record.

1, 2, 3, 6, 20, 40, 80, 1975, 3999, 15995, 39999

Offset: 1

N. J. A. Sloane, Jan 23 2019

The corresponding numbers of steps are 0, 1, 11, 12, 13, 14, 15, 16, 17.

			a(4)=20 refers to the fact that it takes 13 steps to reach 100 from 5 using the Choix de Bruxelles (version 2) operation, and all multiples of 5 less than 100 can be reached from 5 in fewer than 13 steps.

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, 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.

Cf. A323286, A323460, A323454, A323484.

a(10) from Michael S. Branicky, Oct 05 2024

a(11) from Michael S. Branicky, Oct 11 2024

A356511 Total number of distinct numbers that can be obtained by starting with 1 and applying the "Choix de Bruxelles", version 2 operation at most n times in duodecimal (base 12).

Original entry on oeis.org

1, 2, 3, 4, 5, 9, 19, 45, 107, 275, 778, 2581, 10170, 45237, 222859, 1191214, 6887258, 42894933, 287397837

Offset: 0

J. Conrad, Aug 09 2022

			For n=4, the a(4) = 5 numbers obtained are (in base 12): 1, 2, 4, 8, 14.
For n=5, they expand to a(5) = 9 numbers (in base 12): 1, 2, 4, 8, 12, 14, 18, 24, 28.

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.
J. Conrad, Python program.

Cf. A323289 (decimal).

Python
```
# See Conrad link.
```

from itertools import islice
from sympy.ntheory import digits
def fd12(d): return sum(12**i*di for i, di in enumerate(d[::-1]))
def cdb2(n):
    d, out = digits(n, 12)[1:], {n}
    for l in range(1, len(d)+1):
        for i in range(len(d)+1-l):
            if d[i] == 0: continue
            t = fd12(d[i:i+l])
            out.add(fd12(d[:i] + digits(2*t, 12)[1:] + d[i+l:]))
            if t&1 == 0:
                out.add(fd12(d[:i] + digits(t//2, 12)[1:] + d[i+l:]))
    return out
def agen():
    reach, expand = {1}, [1]
    while True:
        yield len(reach)
        newreach = {r for q in expand for r in cdb2(q) if r not in reach}
        reach |= newreach
        expand = list(newreach)
print(list(islice(agen(), 14))) # Michael S. Branicky, Aug 17 2022

a(16)-a(18) from Michael S. Branicky, Aug 17 2022

A360703 Starting from 1, successively take the smallest "Choix de Bruxelles" with factor 3 which is not already in the sequence.

Original entry on oeis.org

1, 3, 9, 27, 67, 187, 129, 43, 41, 121, 17, 37, 97, 277, 677, 1877, 1297, 199, 133, 111, 113, 119, 139, 339, 313, 311, 331, 131, 191, 193, 393, 333, 399, 999, 933, 911, 913, 919, 319, 357, 157, 57, 19, 13, 11, 31, 33, 39, 99, 93, 91, 271, 273, 279, 679, 673, 671, 1871, 1291, 197, 137, 117, 151, 51, 53, 59, 159, 153

Offset: 0

Alon Vinkler, Feb 16 2023

At a given term t, the Choix de Bruxelles with factor 3 can choose to multiply any decimal digit substring (not starting 0) of t by 3 or divide by 3 if that substring is divisible by 3.
These choices on substrings give various possible next values and here take the smallest not yet in the sequence.
The sequence can be finite if the only choices we have are already in the sequence, but this has not been found in the first 1125299 terms.

			Below, square brackets [] represent multiplication by 3(e.g., [4] = 12); curly brackets {} represent division by 3 (e.g., {6} = 2); digits outside the brackets are not affected by the multiplication or division (e.g., 1[3] = 19 and 1{18} = 16).
We begin with 1 and, at each step, we go to the smallest number possible that hasn't yet appeared in the sequence:
 1 --> [1] = 3
 3 --> [3] = 9
 9 --> [9]= 27
 27--> [2]7 = 67
 67--> [6]7= 187
 187 --> 1{87}=129
 129 --> {129} = 43
 ... and so on.

Alon Vinkler, Table of n, a(n) for n = 0..10000
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.
Alon Vinkler, C# Program

Cf. A323286, A358708, A360190.

A323462 Smallest number that can be obtained from the "Choix de Bruxelles", version 2 (A323460) operation applied to n.

Original entry on oeis.org

1, 1, 3, 2, 5, 3, 7, 4, 9, 5, 11, 6, 13, 7, 15, 8, 17, 9, 19, 10, 11, 11, 13, 12, 15, 13, 17, 14, 19, 15, 31, 16, 33, 17, 35, 18, 37, 19, 39, 20, 21, 21, 23, 22, 25, 23, 27, 24, 29, 25, 51, 26, 53, 27, 55, 28, 57, 29, 59, 30, 31, 31, 33, 32, 35, 33, 37, 34, 39, 35, 71, 36, 73, 37, 75, 38, 77, 39, 79, 40, 41

Offset: 1

N. J. A. Sloane, Jan 23 2019

Smallest element in row n of irregular triangle in A323460.

Theorem: Let the decimal expansion of n be d_1 d_2 ... d_k. (i) If there is a substring d_r ... d_s which starts with d_r = 1 and ends with an even digit d_s = e, take that string which starts with the leftmost 1 and ends with the rightmost even digit, and halve it. (ii) Otherwise, if there is an even digit e, take the substring from d_1 to the rightmost such e and halve it. (iii) Otherwise, all d_i are odd, and a(n) = n.

			From 23 we can reach any of 13, 43, 26, 46, and the smallest of these is a(23) = 13.

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, 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.

Cf. A323286, A323460, A323288.

A323463 Values of n at which A323454 reaches a new record.

Original entry on oeis.org

1, 2, 3, 99, 369, 999, 1999, 9879, 19979

Offset: 1

N. J. A. Sloane, Jan 23 2019

The corresponding numbers of steps are 0, 1, 11, 12, 13, 14, 15, 16.

			a(4)=99 refers to the fact that it takes 12 steps to reach 99 from 1 using the Choix de Bruxelles (version 2) operation, and all numbers less than 99 (and not ending in 0 or 5) can be reached from 1 in fewer than 12 steps.

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.

Cf. A323286, A343460, A323454.

a(9) from Michael S. Branicky, Oct 01 2024

A334629 Smallest number that can be obtained by starting with 1 and applying "Choix de Bruxelles (version 2)" (see A323460) n times without backtracking or repeating.

Original entry on oeis.org

1, 2, 4, 8, 16, 13, 23, 17, 14, 7, 6, 3, 99, 369, 999, 1999, 9879, 19979

Offset: 0

Philip C. Ritchey, Sep 09 2020

			a(0)-a(4) = 1,2,4,8,16 by applying 0,1,2,3,4 steps: 1->2->4->8->16.
a(5) = 13 by applying 5 steps: 1->2->4->8->16->13 (halve the 6 in 16).
a(11) = 3 by applying 11 steps to reach 3 from 1.

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.

Cf. A323454, A323460.

A323454(a(n)) = n by definition.

a(17) from Michael S. Branicky, Oct 01 2024

A356715 Total number of distinct numbers that can be obtained by starting with 1 and applying the "Choix de Bruxelles", version 2 operation at most n times in ternary (base 3).

Original entry on oeis.org

1, 2, 3, 6, 11, 26, 68, 177, 492, 1403, 4113, 12149, 36225, 108268, 324529, 973163, 2920533, 8764041, 26303715, 78935398, 236878491, 710783343

Offset: 0

J. Conrad, Aug 24 2022

			For n = 2, a(2) = 3 since the numbers obtained are (in base 3): 1, 2, 11.
For n = 4, they expand to a(5) = 11 numbers (in base 3): 1, 2, 11, 12, 21, 22, 101, 111, 112, 121, 211.

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.
J. Conrad, Python program.

Cf. A323289 (decimal), A356511 (base 12)

Python
```
# See Conrad link.
```

from itertools import islice
from sympy.ntheory import digits
def fd(d, b): return sum(b**i*di for i, di in enumerate(d[::-1]))
def cdb2(n, base=3):
    d, out = digits(n, base)[1:], {n}
    for l in range(1, len(d)+1):
        for i in range(len(d)+1-l):
            if d[i] == 0: continue
            t = fd(d[i:i+l], base)
            out.add(fd(d[:i] + digits(2*t, base)[1:] + d[i+l:], base))
            if t&1 == 0:
                out.add(fd(d[:i] + digits(t//2, base)[1:] + d[i+l:], base))
    return out
def agen():
    reach, expand = {1}, [1]
    while True:
        yield len(reach) #; print(reach); print([digits(t, 3)[1:] for t in sorted(reach)])
        newreach = {r for q in expand for r in cdb2(q) if r not in reach}
        reach |= newreach
        expand = list(newreach)
print(list(islice(agen(), 10))) # Michael S. Branicky, Aug 24 2022

a(15)-a(19) from Michael S. Branicky, Aug 24 2022

a(20)-a(21) from Michael S. Branicky, Aug 30 2022

cp's OEIS Frontend

A323464 Values of n at which A323484 reaches a new record.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

A356511 Total number of distinct numbers that can be obtained by starting with 1 and applying the "Choix de Bruxelles", version 2 operation at most n times in duodecimal (base 12).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Python

Python

Extensions

A360703 Starting from 1, successively take the smallest "Choix de Bruxelles" with factor 3 which is not already in the sequence.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A323462 Smallest number that can be obtained from the "Choix de Bruxelles", version 2 (A323460) operation applied to n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A323463 Values of n at which A323454 reaches a new record.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

A334629 Smallest number that can be obtained by starting with 1 and applying "Choix de Bruxelles (version 2)" (see A323460) n times without backtracking or repeating.

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

Extensions

A356715 Total number of distinct numbers that can be obtained by starting with 1 and applying the "Choix de Bruxelles", version 2 operation at most n times in ternary (base 3).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Python

Python

Extensions