A367288 Lexicographically earliest sequence of distinct nonnegative integers such that for any n > 0, a(n-1) and a(n) are congruent modulo n, and the least value not yet in the sequence appears as soon as possible.

0, 1, 5, 2, 18, 3, 39, 4, 60, 6, 106, 7, 151, 8, 204, 9, 265, 10, 334, 11, 411, 12, 496, 13, 589, 14, 690, 15, 799, 16, 916, 17, 1009, 19, 1175, 20, 1316, 21, 1465, 22, 1622, 23, 1787, 24, 1960, 25, 2141, 26, 2330, 27, 2527, 28, 2732, 29, 2945, 30, 3166, 31

Offset: 0

Author

Rémy Sigrist, Nov 12 2023

Keywords

• nonn

Comments

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) and v are congruent modulo 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).

This construction is similar to that of A352713.

This sequence is a variant of A125717 and, by design, is guaranteed to be a permutation of the nonnegative integers (with inverse A367289).

Examples

			The first terms are:
  n   a(n)  a(n-1) mod n  a(n) mod n
  --  ----  ------------  ----------
   0     0  N/A           N/A
   1     1             0           0
   2     5             1           1
   3     2             2           2
   4    18             2           2
   5     3             3           3
   6    39             3           3
   7     4             4           4
   8    60             4           4
   9     6             6           6
  10   106             6           6
  11     7             7           7
  12   151             7           7
  13     8             8           8
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 0..10000
• Rémy Sigrist, PARI program
• Index entries for sequences related to Recamán's sequence
• Index entries for sequences that are permutations of the natural numbers

Crossrefs

Cf. A005132, A125717, A352713, A367289 (inverse), A367290.

Programs

PARI

See Links section.