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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.

%I A371220 #12 Mar 16 2024 17:26:36
%S A371220 1,3,2,10,4,52,5,43,6,54,7,65,8,76,9,103,11,99,12,110,13,121,14,132,
%T A371220 15,143,16,154,17,165,18,176,19,187,20,198,21,209,22,220,23,231,24,
%U A371220 242,25,253,26,264,27,275,28,286,29,297,30,308,31,319,32,330,33,341
%N 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.
%C A371220 To build the sequence:
%C A371220 - we start with a(1) = 1,
%C A371220 - once we have n terms, let v be the least value not yet in the sequence,
%C A371220 - if a(1) + ... a(n) + v is a multiple of n+1, then a(n+1) = v,
%C A371220 - 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).
%C A371220 The construction is similar to that of A367288.
%C A371220 This sequence is a variant of A019444 and, by design, is guaranteed to be a permutation of the nonnegative integers (with inverse A371221).
%H A371220 Rémy Sigrist, <a href="/A371220/b371220.txt">Table of n, a(n) for n = 1..10000</a>
%H A371220 Rémy Sigrist, <a href="/A371220/a371220.gp.txt">PARI program</a>
%H A371220 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%e A371220 The first terms, alongside the average of the first terms, are:
%e A371220   n   a(n)  (a(1)+...+a(n))/n
%e A371220   --  ----  -----------------
%e A371220    1     1                  1
%e A371220    2     3                  2
%e A371220    3     2                  2
%e A371220    4    10                  4
%e A371220    5     4                  4
%e A371220    6    52                 12
%e A371220    7     5                 11
%e A371220    8    43                 15
%e A371220    9     6                 14
%e A371220   10    54                 18
%e A371220   11     7                 17
%e A371220   12    65                 21
%o A371220 (PARI) See Links section.
%Y A371220 Cf. A019444, A367288, A371221 (inverse).
%K A371220 nonn
%O A371220 1,2
%A A371220 _Rémy Sigrist_, Mar 15 2024