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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).
8
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, 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, 33, 14, 34, 15, 35, 36, 3, 37, 16, 38, 39, 40, 7, 41, 42, 17, 43, 18, 44, 45, 8
OFFSET
1,3
COMMENTS
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.
Conjecture: For all k > 0, there is exactly one j such that a(j) = a(j+k). For 0 < k < 11911, this conjecture holds.
LINKS
EXAMPLE
1 1 2 1 3 4 2
(1)1 2 1 3 4 k = 1
1(1)2 1 3 k = 2
(1)1 2 1 k = 3
1 1(2) k = 4
1 1 k = 5
1 k = 6
Coincidences are circled. There can only be one coincidence per row.
a(3) cannot be 1 because that would result in two coincidences for k = 1.
a(5) cannot be 1 or 2 because those values would result in two coincidences for k = 1 and k = 2, respectively.
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.
PROG
(Python) # See Links section.
CROSSREFS
KEYWORD
nonn,nice
AUTHOR
Samuel B. Reid, Aug 19 2020
STATUS
approved