A345145 Lexicographically earliest sequence of nonnegative terms forming a clockwise square spiral when nothing else is read except the prime/nonprime nature of the terms or the prime/nonprime nature of the digits (see the Comments section).

0, 2, 3, 1, 5, 4, 7, 6, 8, 29, 9, 20, 11, 21, 10, 24, 19, 12, 14, 16, 31, 18, 40, 26, 44, 41, 28, 46, 48, 15, 61, 49, 60, 30, 64, 66, 89, 34, 68, 69, 80, 36, 101, 81, 84, 38, 86, 88, 90, 13, 91, 94, 96, 98, 42, 99, 109, 100, 45, 104, 106, 108, 62, 110, 149, 111, 63, 114, 116, 118, 102, 119, 181, 140, 105

Offset: 1

Author

Eric Angelini and Carole Dubois, Jun 09 2021

Keywords

• base
• nonn

Comments

A nonprime term turns the pencil 0 degree to the right, then advances 1 unit; a prime term turns the pencil 90 degrees to the right, then advances 1 unit.

The same spiral is achieved when reading one by one the nonprime/prime digits of the sequence (instead of the terms) and using the pencil accordingly.

This is possible as the terms of the sequence were carefully chosen to place a prime term (respectively, a prime digit) at every corner of the spiral (and nowhere else).

Examples

			The pencil points towards the East before the start. The 0 doesn't change its orientation and a 1-unit line directed towards the East is traced. As 2 is prime, the pencil turns 90 degrees towards the South and a 1-unit line is traced. As 3 is prime, the pencil turns 90 degrees towards the West and a 1-unit line is traced. As 1 is nonprime, the pencil doesn't change its orientation (towards the West) and a 1-unit line is traced. As 5 is prime, the pencil turns 90 degrees towards the North and a 1-unit line is traced. As 4 is nonprime, the pencil doesn't change its orientation (towards the North) and a 1-unit line is traced. Etc.
When it comes to a(10) = 29, the "term-only-pencil" reads 29 (prime), changes its orientation (by making a right turn) and advances 1 square, then reads a(11) = 9 (nonprime), doesn't change its orientation and advances 1 square;
Similarly, when it comes to a(10) = 29, the "digits-only-pencil" reads the digit 2 (prime), changes its orientation (by making a right turn) and advances 1 square, then reads the digit 9 (nonprime), doesn't change its orientation and advances 1 square;
Those two behaviors of the pencils are equivalent when it comes to draw the clockwise square spiral.
		

Crossrefs

Cf. A344547, A344548, A174344 (an example of a clockwise square spiral).