A323286 -id:A323286 - cp's OEIS frontend

A368194 Irregular table T(n, k), n > 0, k = 1..A368195(n), read by rows: the n-th row lists the numbers that can be obtained by replacing any positive number without leading zeros, say m, appearing in the decimal expansion of n by one of the divisors of m.

1, 1, 2, 1, 3, 1, 2, 4, 1, 5, 1, 2, 3, 6, 1, 7, 1, 2, 4, 8, 1, 3, 9, 1, 2, 5, 10, 1, 11, 1, 2, 3, 4, 6, 11, 12, 1, 11, 13, 1, 2, 7, 11, 12, 14, 1, 3, 5, 11, 15, 1, 2, 4, 8, 11, 12, 13, 16, 1, 11, 17, 1, 2, 3, 6, 9, 11, 12, 14, 18, 1, 11, 13, 19, 1, 2, 4, 5, 10, 20

Offset: 1

Rémy Sigrist, Dec 16 2023

The n-th row starts with 1, ends with n, and contains the divisors of n (A027750).

			Table T(n, k) begins:
    1;
    1, 2;
    1, 3;
    1, 2, 4;
    1, 5;
    1, 2, 3, 6;
    1, 7;
    1, 2, 4, 8;
    1, 3, 9;
    1, 2, 5, 10;
    1, 11;
    1, 2, 3, 4, 6, 11, 12;
    1, 11, 13;
    1, 2, 7, 11, 12, 14;
    1, 3, 5, 11, 15;
    1, 2, 4, 8, 11, 12, 13, 16;
    1, 11, 17;
    1, 2, 3, 6, 9, 11, 12, 14, 18;
      ...

Rémy Sigrist, Table of n, a(n) for n = 1..16938 (rows for n = 1..1000 flattened)
Rémy Sigrist, PARI program
Index entries for sequences related to decimal expansion of n

Cf. A027750, A323286, A342072, A368195, A368313 (binary variant).

PARI
```
See Links section.
```

T(n, 1) = 1.

T(n, A368195(n)) = n.

A323452 First differences of A323289.

Original entry on oeis.org

1, 1, 1, 1, 4, 15, 35, 77, 226, 1003, 4627, 22195, 107764, 553107, 3219398, 20940662, 146173751, 1077052928

Offset: 1

N. J. A. Sloane, Jan 15 2019

Rémy Sigrist, C# program

Cf. A323286-A323289.

a(7)-a(16) from Rémy Sigrist, Jan 15 2019
a(17) from Michael S. Branicky, Jul 24 2022
a(18) from Michael S. Branicky, Jul 26 2022

A323456 Irregular triangle read by rows: row n lists the numbers that can be obtained from the binary expansion of n by either deleting a single 0, or inserting a single 0 after any 1.

Original entry on oeis.org

2, 1, 4, 5, 6, 2, 8, 3, 9, 10, 3, 10, 12, 11, 13, 14, 4, 16, 5, 17, 18, 5, 6, 18, 20, 7, 19, 21, 22, 6, 20, 24, 7, 21, 25, 26, 7, 22, 26, 28, 23, 27, 29, 30, 8, 32, 9, 33, 34, 9, 10, 34, 36, 11, 35, 37, 38, 10, 12, 36, 40, 11, 13, 37, 41, 42, 11, 14, 38, 42, 44

Offset: 1

N. J. A. Sloane, Jan 17 2019

All the numbers in row n have the same binary weight (A000120) as n.
If k appears in row n, n appears in row k.
If we form a graph on the positive integers by joining k to n if k appears in row n, then there is a connected component for each weight 1, 2, , ...
The smallest number in the component containing n is 2^A000120(n)-1, and n is reachable from 2^A000120(n)-1 in A023416(n) steps. - Rémy Sigrist, Jan 17 2019

			From 6 = 110 we can get 11 = 3, 1010 = 10, or 1100 = 12, so row 6 is {3,10,12}.
From 7 = 111 we can get 1011 = 11, 1101 = 13, or 1110 = 14, so row 7 is {11,13,14}.
The triangle begins:
  2,
  1, 4,
  5, 6,
  2, 8,
  3, 9, 10,
  3, 10, 12,
  11, 13, 14,
  4, 16,
  5, 17, 18,
  5, 6, 18, 20,
  7, 19, 21, 22,
  ...

Rémy Sigrist, Rows n = 1..1000, flattened

Cf. A000120, A323455, A323465.

This is a base-2 analog of A323286.

r323456[n_] := Module[{digs=IntegerDigits[n, 2]} ,Map[FromDigits[#, 2]&, Union[Map[Insert[digs, 0, #+1]&, Flatten[Position[digs, 1]]], Map[Drop[digs, {#}]&, Flatten[Position[digs, 0]]]]]] (* nth row *)
a323456[{m_, n_}] := Flatten[Map[r323456, Range[m, n]]]
a323456[{1,22}] (* Hartmut F. W. Hoft, Oct 24 2023 *)

row(n) = { my (r=Set(), w=0, s=0); while (n, my (v=1+valuation(n,2)); r = setunion(r, Set(n*2^(w+1)+s)); if (v>1, r = setunion(r, Set(n*2^(w-1)+s))); s += (n%(2^v))*2^w; w += v; n \= 2^v); r } \\ Rémy Sigrist, Jan 27 2019

def row(n):
    b = bin(n)[2:]
    s1 = set(b[:i+1] + "0" + b[i+1:] for i in range(len(b)) if b[i] == "1")
    s2 = set(b[:i] + b[i+1:] for i in range(len(b)) if b[i] == "0")
    return sorted(int(w, 2) for w in s1 | s2)
print([c for n in range(1, 23) for c in row(n)]) # Michael S. Branicky, Jul 24 2022

More terms from Rémy Sigrist, Jan 27 2019

A323484 Minimal number of steps to reach 5*n from 5 using "Choix de Bruxelles" version 2 (cf. A323460).

Original entry on oeis.org

0, 1, 11, 2, 11, 12, 8, 3, 10, 12, 9, 11, 7, 9, 10, 4, 9, 10, 10, 13, 10, 10, 7, 10, 8, 6, 9, 9, 9, 9, 8, 5, 10, 8, 11, 9, 10, 9, 11, 14, 11, 11, 8, 10, 8, 7, 10, 9, 9, 9, 9, 6, 9, 9, 11, 8, 9, 10, 10, 10, 9, 7, 12, 6, 11, 11, 11, 7, 11, 12, 10, 10, 11, 10, 13

Offset: 1

Rémy Sigrist, Jan 16 2019

This is equally the minimal number of steps to reach 5*n from 5 using "Choix de Bruxelles" version 1 (cf. A323286). - N. J. A. Sloane, Jan 24 2019

The sequence is well defined as we can reach any multiple of 5 starting from 5.

Rémy Sigrist, Table of n, a(n) for n = 1..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, 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)

Cf. A323286, A323454, A323460, A323464.

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).

Original entry on oeis.org

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

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

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

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

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

PARI
```
See Links section.
```

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

A360190 Starting from 1, successively take the smallest "Choix de Bruxelles" with factor 13 which is not already in the sequence.

Original entry on oeis.org

1, 13, 133, 1333, 13333, 133333, 1333333, 125641, 1256413, 12564133, 1197241, 117481, 9037, 90391, 9031, 90313, 903133, 90241, 902413, 9024133, 90241333, 6941641, 693241, 6932413, 69324133, 6717241, 671557, 65557, 5557, 55591, 5431, 54313, 543133, 54241

Offset: 0

Alon Vinkler, Jan 29 2023

At a given term t, the Choix de Bruxelles with factor 13 can choose to multiply any decimal digit substring (not starting 0) of t by 13, or divide by 13 if that substring is divisible by 13.
These choices on substrings give various possible next values and here take the smallest not yet in the sequence.
The sequence is finite and ends at a(6851) = 7, since the sole next Choix there is multiplication by 13 to 91, but 91 is already in the sequence at the preceding a(6850) = 91.

			Below, square brackets [] represent multiplication by 13 (e.g., [4] = 52); curly brackets {} represent division by 13 (e.g., {26} = 2); digits outside the brackets are not affected by the multiplication or division (e.g., 1[3] = 139 and 1{169} = 113).
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] = 13
 13 --> [1]3 = 133
 133 --> [1]33 = 1333
 1333 --> [1]333 = 13333
 13333 --> [1]3333 = 133333
 133333 --> [1]33333 = 1333333
 1333333 --> 1{333333} = 125641
 ... and so on.

Alon Vinkler, Table of n, a(n) for n = 0..6851
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. A358708 (steps by factor 2), A323286 (Choix with factor 2).

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

Original entry on oeis.org

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

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

cp's OEIS Frontend

A368194 Irregular table T(n, k), n > 0, k = 1..A368195(n), read by rows: the n-th row lists the numbers that can be obtained by replacing any positive number without leading zeros, say m, appearing in the decimal expansion of n by one of the divisors of m.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A323452 First differences of A323289.

Views

Author

Keywords

Links

Crossrefs

Extensions

A323456 Irregular triangle read by rows: row n lists the numbers that can be obtained from the binary expansion of n by either deleting a single 0, or inserting a single 0 after any 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

Extensions

A323484 Minimal number of steps to reach 5*n from 5 using "Choix de Bruxelles" version 2 (cf. A323460).

Views

Author

Keywords

Comments

Links

Crossrefs

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).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

A360190 Starting from 1, successively take the smallest "Choix de Bruxelles" with factor 13 which is not already in the sequence.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

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