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.

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

Author

Rémy Sigrist, Mar 15 2024

Keywords

• nonn

Comments

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).

Examples

			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
		

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. A019444, A367288, A371221 (inverse).

Programs

PARI

See Links section.