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.

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

Author

Rémy Sigrist, Nov 12 2023

Keywords

• nonn

Comments

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).

Examples

			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
		

Links

• 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

Crossrefs

Cf. A364054, A367288, A367291 (inverse).

Programs

PARI

See Links section.