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A337226 Lexicographically earliest sequence of positive integers with the property that, for all k > 0, there is at most one j such that a(j) = a(j+k).

%I A337226 #24 May 24 2021 00:08:49
%S A337226 1,1,2,1,3,4,2,5,1,6,3,7,8,9,4,10,2,11,5,12,1,13,6,14,15,3,16,7,17,18,
%T A337226 8,19,20,21,22,9,23,4,24,10,25,2,26,11,27,5,28,12,29,1,30,13,31,6,32,
%U A337226 33,14,34,15,35,36,3,37,16,38,39,40,7,41,42,17,43,18,44,45,8
%N A337226 Lexicographically earliest sequence of positive integers with the property that, for all k > 0, there is at most one j such that a(j) = a(j+k).
%C A337226 The sequence initially appears to be trivially fractal in that the removal of the first occurrence of each value seems to yield the original sequence. This pattern continues until a(121) where, if the sequence were fractal in this way, the value would be 72 or 1. The actual value is 13, so the pattern is broken.
%C A337226 Conjecture: For all k > 0, there is exactly one j such that a(j) = a(j+k). For 0 < k < 11911, this conjecture holds.
%H A337226 Samuel B. Reid, <a href="/A337226/b337226.txt">Table of n, a(n) for n = 1..10000</a>
%H A337226 Samuel B. Reid, <a href="/A337226/a337226.py.txt">Python program for A337226</a>
%e A337226   1 1 2 1 3 4 2
%e A337226    (1)1 2 1 3 4   k = 1
%e A337226       1(1)2 1 3   k = 2
%e A337226        (1)1 2 1   k = 3
%e A337226           1 1(2)  k = 4
%e A337226             1 1   k = 5
%e A337226               1   k = 6
%e A337226 Coincidences are circled. There can only be one coincidence per row.
%e A337226 a(3) cannot be 1 because that would result in two coincidences for k = 1.
%e A337226 a(5) cannot be 1 or 2 because those values would result in two coincidences for k = 1 and k = 2, respectively.
%e A337226 a(7) cannot be 1, 3, or 4 because those values would result in two coincidences for k = 3, k = 2, and k = 1, respectively. It can, however, be 2 because this results in no double coincidences.
%o A337226 (Python) # See Links section.
%Y A337226 Cf. A003602, A014552, A026272.
%K A337226 nonn,nice
%O A337226 1,3
%A A337226 _Samuel B. Reid_, Aug 19 2020