A262323 -id:A262323 - cp's OEIS frontend

A262255 Inverse permutation to A262323.

1, 5, 10, 16, 20, 24, 28, 32, 36, 2, 3, 4, 9, 15, 21, 23, 29, 31, 37, 6, 8, 7, 11, 17, 19, 25, 27, 33, 35, 12, 14, 18, 13, 41, 46, 39, 48, 50, 52, 42, 22, 26, 40, 43, 44, 56, 64, 54, 66, 45, 30, 34, 47, 55, 59, 58, 60, 62, 80, 57, 38, 70, 49, 65, 61, 69, 68

Offset: 1

Paul Tek, Sep 19 2015

Paul Tek, Table of n, a(n) for n = 1..10000

Cf. A262323.

Cf. A262367 (fixed points).

import Data.List (elemIndex); import Data.Maybe (fromJust)
a262255 = (+ 1) . fromJust . (`elemIndex` a262323_list)
-- Reinhard Zumkeller, Sep 21 2015

A262367 Fixed points of permutations A262323 and A262255.

Original entry on oeis.org

1, 27, 86, 111, 272, 547, 608, 687, 808, 929, 3890, 5557, 7180, 21659, 21663, 60486, 71074, 279428, 603224, 603228, 610798, 710551, 811026, 930320

Offset: 1

Reinhard Zumkeller, Sep 21 2015

Cf. A262255, A262323.

a262367 n = a262367_list !! (n-1)
a262367_list = [x | x <- [1..], a262323 x == x]

a(17)-a(24) from Amiram Eldar, May 11 2024

A285687 Lexicographically earliest sequence of distinct positive terms such that, for any n>0, n and a(n) differ and overlap in base 10 (in the sense of A262323).

Original entry on oeis.org

10, 12, 13, 14, 15, 16, 17, 18, 19, 1, 21, 2, 3, 4, 5, 6, 7, 8, 9, 22, 11, 20, 30, 32, 42, 52, 62, 72, 82, 23, 33, 24, 31, 40, 43, 53, 63, 73, 83, 34, 44, 25, 35, 41, 50, 54, 64, 74, 84, 45, 55, 26, 36, 46, 51, 60, 65, 75, 85, 56, 66, 27, 37, 47, 57, 61, 70

Offset: 1

Rémy Sigrist, May 03 2017

The definition of the overlapping of two numbers is that of the sequence A262323.

This sequence is a self-inverse permutation of the natural numbers.

			The first terms are:
n:    1,   2,  3,  4,  5,  6,  7,  8,  9, 10,  11, 12, 13, 14, 15,...
a(n): 10, 12, 13, 14, 15, 16, 17, 18, 19, 1,  21,   2,  3,  4,  5,...

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PERL program for A285687
Index entries for sequences that are permutations of the natural numbers

Cf. A262323.

Expanded definition. - N. J. A. Sloane, Oct 29 2023

A262411 Lexicographically earliest sequence of distinct terms such that the ternary representations of two consecutive terms overlap.

Original entry on oeis.org

1, 3, 4, 5, 2, 6, 8, 7, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 19, 23, 21, 25, 22, 26, 24, 29, 28, 27, 30, 31, 32, 34, 33, 37, 35, 38, 39, 36, 40, 41, 42, 43, 44, 46, 45, 49, 47, 50, 48, 52, 51, 54, 53, 55, 56, 57, 59, 58, 60, 61, 62, 63, 65, 64, 68, 66

Offset: 1

Reinhard Zumkeller, Sep 22 2015

Suggested by Paul Tek's A262323;
two numbers are overlapping if a nonempty prefix of one equals a suffix of the other;
permutation of the natural numbers with inverse A262429;
A262412(n) = A007089(a(n)).

			.   n | a(n) | A262412(n)           n | a(n) | A262412(n)
. ----+------+-----------         ----+------+-------------
.                                 (25 |   26 |         222 )
.   1 |    1 |   1                 26 |   24 |          220
.   2 |    3 |   10                27 |   29 |       1002
.   3 |    4 |  11                 28 |   28 |    1001
.   4 |    5 |   12                29 |   27 |       1000
.   5 |    2 |    2                30 |   30 |     1010
.   6 |    6 |    20               31 |   31 |  1011
.   7 |    8 |   22                32 |   32 |     1012
.   8 |    7 |    21               33 |   34 |  1021
.   9 |    9 |     100             34 |   33 |     1020
.  10 |   10 |   101               35 |   37 |  1101
.  11 |   11 |     102             36 |   35 |     1022
.  12 |   12 |    110              37 |   38 |    1102
.  13 |   13 |   111               38 |   39 |   1110
.  14 |   14 |    112              39 |   36 |    1100
.  15 |   15 |     120             40 |   40 |  1111
.  16 |   16 |   121               41 |   41 |   1112
.  17 |   17 |     122             42 |   42 |    1120
.  18 |   18 |       200           43 |   43 | 1121
.  19 |   20 |     202             44 |   44 |    1122
.  20 |   19 |       201           45 |   46 | 1201
.  21 |   23 |     212             46 |   45 |    1200
.  22 |   21 |       210           47 |   49 | 1211
.  23 |   25 |      221            48 |   47 |    1202
.  24 |   22 |       211           49 |   50 |  1212
.  25 |   26 |     222             50 |   48 |    1210  .
. (26 |   24 |      220 )

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for sequences that are permutations of the natural numbers

Cf. A262323, A030341, A007089, A262412 (ternary conversion), A262429 (inverse), A262435 (fixed points).

Cf. A262460.

import Data.List (inits, tails, intersect, delete, genericIndex)
a262411 n = genericIndex a262411_list (n - 1)
a262411_list = 1 : f [1] (drop 2 a030341_tabf) where
   f xs tss = g tss where
     g (ys:yss) | null (intersect its $ tail $ inits ys) &&
                  null (intersect tis $ init $ tails ys) = g yss
                | otherwise = (foldr (\t v -> 3 * v + t) 0 ys) :
                              f ys (delete ys tss)
     its = init $ tails xs; tis = tail $ inits xs

A262460 Lexicographically earliest sequence of distinct terms such that the hexadecimal representations of two consecutive terms overlap.

Original entry on oeis.org

1, 16, 17, 18, 2, 32, 34, 33, 19, 3, 35, 48, 51, 49, 20, 4, 36, 50, 37, 5, 21, 65, 22, 6, 38, 66, 39, 7, 23, 81, 24, 8, 40, 82, 41, 9, 25, 97, 26, 10, 42, 98, 43, 11, 27, 113, 28, 12, 44, 114, 45, 13, 29, 129, 30, 14, 46, 130, 47, 15, 31, 145, 57, 67, 52, 64

Offset: 1

Reinhard Zumkeller, Sep 23 2015

Suggested by Paul Tek's A262323;
two numbers are overlapping if a nonempty prefix of one equals a suffix of the other;
permutation of the natural numbers with inverse A262461.

			Table of initial terms: the HEX column gives the hexadecimal representation with aligned overlapping digits.
.   n | a(n) | HEX          n | a(n) | HEX          n | a(n) | HEX
. ----+------+-------     ----+------+-------     ----+------+-------
.   1 |    1 |  1          25 |   38 |   26        49 |   44 |    2C
.   2 |   16 |  10         26 |   66 |  42         50 |  114 |   72
.   3 |   17 | 11          27 |   39 |   27        51 |   45 |    2D
.   4 |   18 |  12         28 |    7 |    7        52 |   13 |     D
.   5 |    2 |   2         29 |   23 |   17        53 |   29 |    1D
.   6 |   32 |   20        30 |   81 |  51         54 |  129 |   81
.   7 |   34 |  22         31 |   24 |   18        55 |   30 |    1E
.   8 |   33 |   21        32 |    8 |    8        56 |   14 |     E
.   9 |   19 |    13       33 |   40 |   28        57 |   46 |    2E
.  10 |    3 |     3       34 |   82 |  52         58 |  130 |   82
.  11 |   35 |    23       35 |   41 |   29        59 |   47 |    2F
.  12 |   48 |     30      36 |    9 |    9        60 |   15 |     F
.  13 |   51 |    33       37 |   25 |   19        61 |   31 |    1F
.  14 |   49 |     31      38 |   97 |  61         62 |  145 |   91
.  15 |   20 |      14     39 |   26 |   1A        63 |   57 |  39
.  16 |    4 |       4     40 |   10 |    A        64 |   67 | 43
.  17 |   36 |      24     41 |   42 |   2A        65 |   52 |  34
.  18 |   50 |     32      42 |   98 |  62         66 |   64 |   40
.  19 |   37 |      25     43 |   43 |   2B        67 |   68 |  44
.  20 |    5 |       5     44 |   11 |    B        68 |   69 |   45
.  21 |   21 |      15     45 |   27 |   1B        69 |   80 |    50
.  22 |   65 |     41      46 |  113 |  71         70 |   53 |   35
.  23 |   22 |      16     47 |   28 |   1C        71 |   83 |    53
.  24 |    6 |       6     48 |   12 |    C        72 |   54 |     36

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Hexadecimal
Wikipedia, Hexadecimal
Index entries for sequences that are permutations of the natural numbers

Cf. A262323, A262411, A262437, A262461 (inverse).

import Data.List (inits, tails, intersect, delete, genericIndex)
a262460 n = genericIndex a262460_list (n - 1)
a262460_list = 1 : f [1] (drop 2 a262437_tabf) where
   f xs tss = g tss where
     g (ys:yss) | null (intersect its $ tail $ inits ys) &&
                  null (intersect tis $ init $ tails ys) = g yss
                | otherwise = (foldr (\t v -> 16 * v + t) 0 ys) :
                              f ys (delete ys tss)
     its = init $ tails xs; tis = tail $ inits xs

A333722 Lexicographically earliest permutation of the positive integers such that a(n), a(n+1) and the product a(n)*a(n+1) have in common at least one identical substring.

Original entry on oeis.org

1, 10, 11, 12, 2, 21, 15, 5, 25, 29, 28, 24, 22, 26, 6, 16, 36, 37, 39, 34, 14, 13, 3, 31, 23, 27, 71, 7, 97, 69, 56, 45, 35, 38, 18, 48, 8, 81, 17, 47, 42, 44, 41, 4, 46, 40, 20, 30, 50, 51, 52, 53, 55, 57, 65, 54, 49, 19, 61, 60, 66, 76, 64, 62, 63, 96, 67, 68, 85, 59, 75, 58, 83, 33, 93, 43, 32, 72, 92, 98, 80, 70, 90, 91, 9

Offset: 1

Eric Angelini and Jean-Marc Falcoz, Apr 03 2020

			a(1) = 1 and a(2) = 10 share with their product 10 the substring 1;
a(2) = 10 and a(3) = 11 share with their product 110 the substring 1;
a(3) = 11 and a(4) = 12 share with their product 132 the substring 1;
a(4) = 12 and a(5) = 2 share with their product 24 the substring 2;
a(5) = 2 and a(6) = 21 share with their product 42 the substring 2; etc.

Jean-Marc Falcoz, Table of n, a(n) for n = 1..20002

Cf. A333723 (lists the products a(n) * a(n+1) in their order of appearance here), A333724 (lists the biggest substring shared by a(n), a(n+1) and (a(n)*a(n+1)) in their order of appearance here), A262323 (is the lexicographically earliest sequence of distinct terms such that the decimal representations of two consecutive terms overlap).

A335043 Lexicographically earliest sequence of distinct positive integers such that for any n > 0, the decimal representations of a(2n-1) and of a(2n+1) appear as substrings in the decimal representation of a(2*n).

Original entry on oeis.org

1, 12, 2, 23, 3, 34, 4, 45, 5, 56, 6, 67, 7, 78, 8, 89, 9, 109, 10, 110, 11, 113, 13, 130, 30, 330, 33, 331, 31, 314, 14, 140, 40, 240, 24, 224, 22, 220, 20, 320, 32, 321, 21, 215, 15, 150, 50, 250, 25, 251, 51, 351, 35, 352, 52, 526, 26, 260, 60, 160, 16, 161

Offset: 1

Rémy Sigrist, Jun 01 2020

This sequence has similarities with A281978; here we look for substrings, there for divisors.

This sequence has similarities with A262323: in both sequences, consecutive terms overlap.

			The first terms are:
  n   a(n)
  --  ------
   1  1
   2  12
   3   2
   4   23
   5    3
   6    34
   7     4
   8     45
   9      5
  10      56
  11       6

Rémy Sigrist, Table of n, a(n) for n = 1..10001
Rémy Sigrist, Colored logarithmic scatterplot of the first 10001 terms (where the color denotes the parity of n)
Rémy Sigrist, PARI program for A335043

Cf. A262323, A281978.

PARI
```
See Links section.
```

A360470 Lexicographically earliest sequence of distinct positive integers such that for any n > 0, the k rightmost digits of a(n+1) equal the k leftmost digits of a(n) for some k > 0.

Original entry on oeis.org

1, 11, 21, 2, 12, 31, 3, 13, 41, 4, 14, 51, 5, 15, 61, 6, 16, 71, 7, 17, 81, 8, 18, 91, 9, 19, 101, 10, 110, 111, 121, 112, 131, 113, 141, 114, 151, 115, 161, 116, 171, 117, 181, 118, 191, 119, 201, 20, 22, 32, 23, 42, 24, 52, 25, 62, 26, 72, 27, 82, 28, 92

Offset: 1

Rémy Sigrist, Feb 08 2023

Leading zeros are ignored.
This sequence is a permutation of the positive integers with inverse A360472:
- if a(n) < 10^e, then we can extend the sequence with a number of the form a(n) + k * 10^e (with k > 0),
- by the pigeonhole principle, there are infinitely many terms starting with the same nonzero digit, say with d,
- every number of the form 10*k + d (with k >= 0) appears in the sequence,
- any number v can appear after a term of the form v * 10^k + d (with k > 0).

			The first terms are:
  n   a(n)  a(n) aligned
  --  ----  ------------
   1     1             1
   2    11            11
   3    21           21
   4     2           2
   5    12          12
   6    31         31
   7     3         3
   8    13        13
   9    41       41
  10     4       4
  11    14      14
  12    51     51

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, Scatterplot of the first 1000000 terms
Rémy Sigrist, PARI program
Index entries for sequences that are permutations of the natural numbers

Cf. A262323, A360472 (inverse).

PARI
```
See Links section.
```

A381130 a(n) is the smallest prime not yet in the sequence that contains a substring of size 2 from a(n-1); a(1)=11.

Original entry on oeis.org

11, 113, 13, 131, 31, 311, 211, 421, 521, 523, 23, 223, 227, 127, 271, 71, 571, 157, 151, 251, 257, 457, 557, 577, 277, 677, 67, 167, 163, 263, 269, 569, 563, 463, 461, 61, 613, 137, 37, 337, 233, 239, 139, 313, 317, 17, 173, 73, 373, 379, 79, 179, 479, 47

Offset: 1

Enrique Navarrete, Feb 14 2025

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A107801, A262323.

from itertools import count, islice
from sympy import isprime, nextprime
def agen(): # generator of terms
    aset, an, minp = set(), 11, 13
    while True:
        yield an
        aset.add(an)
        s = str(an)
        targets = set(s[i:i+2] for i in range(len(s)-1))
        p = minp
        w = str(p)
        while p in aset or not any(t in w for t in targets):
            p = nextprime(p)
            w = str(p)
        while minp in aset:
            minp = nextprime(minp)
        an = p
print(list(islice(agen(), 54))) # Michael S. Branicky, Apr 15 2025

A262702 Lexicographically earliest sequence of distinct prime numbers such that the decimal representations of two consecutive terms overlap.

Original entry on oeis.org

2, 23, 3, 13, 11, 17, 7, 37, 43, 31, 19, 41, 101, 61, 103, 71, 47, 73, 67, 79, 97, 29, 229, 293, 307, 53, 5, 59, 359, 83, 283, 311, 107, 131, 109, 151, 113, 137, 181, 127, 191, 139, 211, 149, 241, 157, 251, 163, 271, 167, 281, 173, 313, 193, 317, 179, 331, 197

Offset: 1

Paul Tek, Sep 27 2015

Two terms are said to overlap:
- if the decimal representation of one term is contained in the decimal representation of the other term (for example, 23 and 3 overlap),
- or if, for some k>0, the first k decimal digits (without leading zero) of one term correspond to the k last decimal digits of the other term (for example, 317 and 179 overlap).
This is a variation of A262323 around the prime numbers.
Is this a permutation of the prime numbers?

			The first terms of the sequence are:
+----+--------+
| n  | a(n)   |
+----+--------+
|  1 |  2     |
|  2 |  23    |
|  3 |   3    |
|  4 |  13    |
|  5 | 11     |
|  6 |  17    |
|  7 |   7    |
|  8 |  37    |
|  9 | 43     |
| 10 |  31    |
| 11 |   19   |
| 12 |  41    |
| 13 |   101  |
| 14 |  61    |
| 15 |   103  |
| 16 |  71    |
| 17 | 47     |
| 18 |  73    |
| 19 | 67     |
| 20 |  79    |
| 21 |   97   |
| 22 |  29    |
| 23 | 229    |
| 24 |  293   |
| 25 |    307 |
+----+--------+

Paul Tek, Table of n, a(n) for n = 1..35526
Paul Tek, PERL program for this sequence

Cf. A076653, A262323.

Perl
```
See Links section.
```

cp's OEIS Frontend

A262255 Inverse permutation to A262323.

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A262367 Fixed points of permutations A262323 and A262255.

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A285687 Lexicographically earliest sequence of distinct positive terms such that, for any n>0, n and a(n) differ and overlap in base 10 (in the sense of A262323).

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A262411 Lexicographically earliest sequence of distinct terms such that the ternary representations of two consecutive terms overlap.

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A262460 Lexicographically earliest sequence of distinct terms such that the hexadecimal representations of two consecutive terms overlap.

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A333722 Lexicographically earliest permutation of the positive integers such that a(n), a(n+1) and the product a(n)*a(n+1) have in common at least one identical substring.

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A335043 Lexicographically earliest sequence of distinct positive integers such that for any n > 0, the decimal representations of a(2*n-1) and of a(2*n+1) appear as substrings in the decimal representation of a(2*n).

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A360470 Lexicographically earliest sequence of distinct positive integers such that for any n > 0, the k rightmost digits of a(n+1) equal the k leftmost digits of a(n) for some k > 0.

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A381130 a(n) is the smallest prime not yet in the sequence that contains a substring of size 2 from a(n-1); a(1)=11.

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A335043 Lexicographically earliest sequence of distinct positive integers such that for any n > 0, the decimal representations of a(2n-1) and of a(2n+1) appear as substrings in the decimal representation of a(2*n).