A381537 Lexicographically least sequence of natural numbers such that for all arithmetic progressions p, length(p) <= sqrt(max(p)).

1, 4, 5, 8, 9, 10, 12, 15, 16, 17, 18, 22, 23, 24, 25, 26, 28, 29, 30, 31, 33, 35, 36, 37, 38, 39, 40, 42, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 57, 61, 62, 63, 64, 65, 66, 67, 68, 70, 71, 72, 73, 74, 75, 76, 81, 82, 83, 84, 85, 86, 87, 88, 90, 91, 92

Offset: 1

Author

Samuel Harkness, Feb 26 2025

Keywords

• nonn

Comments

Up to a(n) the longest possible arithmetic progression is sqrt(a(n)).

Does the density of this sequence approach 1?

Examples

			1 is in the sequence, as 1 creates the arithmetic progression p = {1}, where length(p) = 1 and sqrt(max(p)) = 1.
For 2: the arithmetic progression p = {1,2} would be created. Here, length(p) = 2, and sqrt(max(p)) = sqrt(2), so length(p) > sqrt(max(p)), thus 2 is not in the sequence. Similarly, 3 is not in the sequence.
For 4: p = {1,4} is the only new arithmetic progression. Here, length(p) = 2, and sqrt(max(p)) = 2, so 4 is in the sequence. Similarly, 5 is in the sequence.
For 6: the arithmetic progression p = {4,5,6} would be created. Here, length(p) = 3, and sqrt(max(p)) = sqrt(6), so length(p) > sqrt(max(p)), thus 6 is not in the sequence.
		

Links

• Samuel Harkness, MATLAB program

Crossrefs

Cf. A281579, A362815.

Programs

MATLAB

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