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).

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, 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, 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

Offset: 1

Author

Scott R. Shannon, Mar 03 2025

Keywords

• nonn

Comments

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.

It is conjectured that the values of n for which a(n) = 1 is given by A092482.

See A381660 for the single value of k for each distinct positive integer, and A381659 for the index where each such integer first appears.

Examples

			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.
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.
a(6) = 1 as this does not create any three terms equalling 1 separated by any value k, so 1 is again chosen.
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.
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.
		

Links

• Scott R. Shannon, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A381659 (index of first appearance), A381660 (k values), A092482 (indices of 1's), A381597, A229037.