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A381658 Lexicographically earliest sequence of positive integers such that for each distinct positive integer t there is only one value of k such that t = a(n) = a(n+k) = a(n+2*k).

%I A381658 #13 Mar 04 2025 07:32:42
%S A381658 1,1,1,2,2,1,1,2,2,2,3,3,3,1,1,3,1,1,2,2,4,2,2,3,3,4,3,3,4,4,5,4,3,5,
%T A381658 5,1,1,5,1,1,4,4,2,2,1,1,2,1,1,5,3,2,2,5,2,2,3,3,4,3,3,4,5,4,5,3,3,4,
%U A381658 6,2,4,6,2,6,4,6,6,5,3,3,4,3,5,4,4,5,5,6,6,4,6,6,7,7,7,8,5,1,1,5,1,1,6,5,5,7,1,1,2,1,1,2,2,3,3,2,3,8,4,6
%N A381658 Lexicographically earliest sequence of positive integers such that for each distinct positive integer t there is only one value of k such that t = a(n) = a(n+k) = a(n+2*k).
%C A381658 In the first 2.5 million terms the only numbers to appear in three consecutive terms are 1 (at n = 1), 2 (at n = 8), 5 (at n = 11), 7 (at n = 93), 8 (at n = 169), and 112 (at n = 96610). It is unknown if more such numbers exist.
%C A381658 It is conjectured that the values of n for which a(n) = 1 is given by A092482.
%C A381658 See A381660 for the single value of k for each distinct positive integer, and A381659 for the index where each such integer first appears.
%H A381658 Scott R. Shannon, <a href="/A381658/b381658.txt">Table of n, a(n) for n = 1..10000</a>
%e A381658 a(1) = a(2) = a(3) = 1. As 1 has now appeared in three terms satisfying a(n) = a(n+k) = a(n+2*k) = 1, with k = 1 in this instance, no other three terms equalling 1 can appear anywhere in the sequence that would satisfy a similar relationship.
%e A381658 a(4) = a(5) = 2 as choosing 1 would create another three terms equalling 1 separated by 1, and three terms equalling 1 separated by 2, namely a(1), a(3), a(5). As neither of those is permitted, the next smallest number 2 is chosen.
%e A381658 a(6) = 1 as this does not create any three terms equalling 1 separated by any value k, so 1 is again chosen.
%e A381658 a(10) = 2 as choosing 1 would create three terms a(2) = a(6) = a(10) = 1 with a difference of 4 which is not permitted. Note that a(9) = a(10) = a(11) = 2, so no other three terms equalling 2 can appear anywhere in the sequence that would satisfy a(n) = a(n+k) = a(n+2*k) = 2.
%e A381658 a(11) = 3 as choosing 1 would create three terms a(3) = a(7) = a(11) = 1 with a difference of 4, while choosing 2 would create a(9) = a(10) = a(11) = 2 with a difference of 1. As neither is permitted the next smallest number 3 is chosen.
%Y A381658 Cf. A381659 (index of first appearance), A381660 (k values), A092482 (indices of 1's), A381597, A229037.
%K A381658 nonn
%O A381658 1,4
%A A381658 _Scott R. Shannon_, Mar 03 2025