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A345145
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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).
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0
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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
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OFFSET
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1,2
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COMMENTS
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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).
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LINKS
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EXAMPLE
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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.
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CROSSREFS
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KEYWORD
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base,nonn
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AUTHOR
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STATUS
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approved
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