A367288 -id:A367288 - cp's OEIS frontend

A367289 Inverse permutation to A367288.

0, 1, 3, 5, 7, 2, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 4, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 6, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 105, 107, 109, 111, 8, 113, 115, 117, 119

Offset: 0

Rémy Sigrist, Nov 12 2023

nonn

			A367288(77) = 42, hence A367288(42) = 77.

Rémy Sigrist, PARI program
Index entries for sequences that are permutations of the natural numbers

Cf. A367288.

PARI
```
See Links section.
```

A367290 Lexicographically earliest sequence of distinct positive integers such that for any n > 0, a(n) and a(n+1) are congruent modulo the n-th prime number, and the least value not yet in the sequence appears as soon as possible.

Original entry on oeis.org

1, 5, 2, 17, 3, 69, 4, 310, 6, 558, 7, 193, 8, 869, 9, 2077, 10, 1780, 11, 3562, 12, 961, 13, 6155, 14, 2439, 15, 8255, 16, 6120, 18, 12464, 19, 9472, 20, 11195, 21, 4260, 22, 24070, 23, 16133, 24, 18360, 25, 19528, 26, 27456, 27, 25905, 28, 46395, 29, 6054

Offset: 1

Rémy Sigrist, Nov 12 2023

nonn

To build the sequence:
- we start with a(1) = 1, and repeatedly:
- let a(n) be the last known term and v the least value not yet in the sequence,
- if a(n) and v are congruent modulo the n-th prime number then a(n+1) = v,
- otherwise a(n+2) = v and a(n+1) is chosen as small as possible in such a way as to satisfy the required congruences (this is always possible as two consecutive prime numbers are coprime).
This sequence is a variant of A364054, and, by design, is guaranteed to be a permutation of the positive integers (with inverse A367291).

			The first terms are:
  n   a(n)   a(n) mod prime(n)  a(n+1) mod prime(n)
  --  -----  -----------------  -------------------
   1      1                  1                    1
   2      5                  2                    2
   3      2                  2                    2
   4     17                  3                    3
   5      3                  3                    3
   6     69                  4                    4
   7      4                  4                    4
   8    310                  6                    6
   9      6                  6                    6
  10    558                  7                    7
  11      7                  7                    7
  12    193                  8                    8
  13      8                  8                    8

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

Cf. A364054, A367288, A367291 (inverse).

PARI
```
See Links section.
```

A371214 Lexicographically earliest sequence of distinct nonnegative integers such that for any n > 0, a(n-1) + a(n) is a multiple of n, and the least value not yet in the sequence appears as soon as possible.

Original entry on oeis.org

0, 1, 7, 2, 22, 3, 45, 4, 76, 5, 115, 6, 18, 8, 216, 9, 279, 10, 350, 11, 429, 12, 516, 13, 611, 14, 714, 15, 825, 16, 944, 17, 47, 19, 1205, 20, 1348, 21, 55, 23, 1657, 24, 1824, 25, 1999, 26, 2182, 27, 2373, 28, 2572, 29, 2779, 30, 2994, 31, 3217, 32, 3448

Offset: 0

Rémy Sigrist, Mar 15 2024

nonn

To build the sequence:
- we start with a(0) = 0, and repeatedly:
- let a(n) be the last known term and v the least value not yet in the sequence,
- if a(n) + v is a multiple of n+1 then a(n+1) = v,
- otherwise a(n+2) = v and a(n+1) is chosen as small as possible in such a way as to satisfy the required congruences (this is always possible as n+1 and n+2 are coprime).
The construction is similar to that of A367288.
This sequence is a variant of A099506 and, by design, is guaranteed to be a permutation of the nonnegative integers (with inverse A371215).

			The first terms are:
  n   a(n)  (a(n-1) + a(n))/n
  --  ----  -----------------
   0     0  N/A
   1     1                  1
   2     7                  4
   3     2                  3
   4    22                  6
   5     3                  5
   6    45                  8
   7     4                  7
   8    76                 10
   9     5                  9
  10   115                 12
  11     6                 11
  12    18                  2

Rémy Sigrist, PARI program
Index entries for sequences that are permutations of the natural numbers

Cf. A099506, A367288, A371215 (inverse).

PARI
```
See Links section.
```

A371220 Lexicographically earliest sequence of distinct positive integers such that for any n > 0, a(1) + ... + a(n) is a multiple of n, and the least value not yet in the sequence appears as soon as possible.

Original entry on oeis.org

1, 3, 2, 10, 4, 52, 5, 43, 6, 54, 7, 65, 8, 76, 9, 103, 11, 99, 12, 110, 13, 121, 14, 132, 15, 143, 16, 154, 17, 165, 18, 176, 19, 187, 20, 198, 21, 209, 22, 220, 23, 231, 24, 242, 25, 253, 26, 264, 27, 275, 28, 286, 29, 297, 30, 308, 31, 319, 32, 330, 33, 341

Offset: 1

Rémy Sigrist, Mar 15 2024

nonn

To build the sequence:
- we start with a(1) = 1,
- once we have n terms, let v be the least value not yet in the sequence,
- if a(1) + ... a(n) + v is a multiple of n+1, then a(n+1) = v,
- otherwise a(n+2) = v and a(n+1) is chosen as small as possible so as to meet the required conditions (this is always possible as n+1 and n+2 are coprime).
The construction is similar to that of A367288.
This sequence is a variant of A019444 and, by design, is guaranteed to be a permutation of the nonnegative integers (with inverse A371221).

			The first terms, alongside the average of the first terms, are:
  n   a(n)  (a(1)+...+a(n))/n
  --  ----  -----------------
   1     1                  1
   2     3                  2
   3     2                  2
   4    10                  4
   5     4                  4
   6    52                 12
   7     5                 11
   8    43                 15
   9     6                 14
  10    54                 18
  11     7                 17
  12    65                 21

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

Cf. A019444, A367288, A371221 (inverse).

PARI
```
See Links section.
```

cp's OEIS Frontend

A367289 Inverse permutation to A367288.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

A367290 Lexicographically earliest sequence of distinct positive integers such that for any n > 0, a(n) and a(n+1) are congruent modulo the n-th prime number, and the least value not yet in the sequence appears as soon as possible.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A371214 Lexicographically earliest sequence of distinct nonnegative integers such that for any n > 0, a(n-1) + a(n) is a multiple of n, and the least value not yet in the sequence appears as soon as possible.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A371220 Lexicographically earliest sequence of distinct positive integers such that for any n > 0, a(1) + ... + a(n) is a multiple of n, and the least value not yet in the sequence appears as soon as possible.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI