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A306807
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An irregular fractal sequence: underline a(n) iff the absolute difference |a(n-1) - a(n)| is prime; all underlined terms rebuild the starting sequence.
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1
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1, 2, 3, 1, 5, 2, 6, 3, 1, 7, 5, 2, 8, 6, 3, 1, 9, 7, 5, 2, 10, 8, 6, 3, 1, 11, 9, 7, 5, 2, 12, 10, 8, 6, 3, 1, 13, 11, 9, 7, 5, 2, 14, 12, 10, 8, 6, 3, 1, 15, 13, 11, 9, 7, 5, 2, 16, 14, 12, 10, 8, 6, 3, 1, 17, 15, 13, 11, 9, 7, 5, 2, 18, 16, 14, 12, 10, 8, 6, 3, 1, 19, 17, 15, 13, 11, 9, 7, 5, 2, 20, 18, 16, 14, 12, 10, 8, 6, 3, 1
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OFFSET
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1,2
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COMMENTS
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The sequence S starts with a(1) = 1 and a(2) = 2. S is extended by duplicating the first term A among the not yet duplicated terms, under the condition that the absolute difference |a(n-1) - a(n)| is prime. If this is not the case, we then extend S with the smallest integer X not yet present in S such that the absolute difference |a(n-1) - a(n)| is not prime. S is the lexicographically earliest sequence with this property.
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LINKS
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EXAMPLE
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S starts with a(1) = 1 and a(2) = 2
Can we duplicate a(1) to form a(3)? No, as |a(2) - a(3)| would be 1 and 1 is not prime. We thus extend S with the smallest integer X not yet in S such that |a(2) - X| is not prime. We get a(3) = 3.
Can we duplicate a(1) to form a(4)? Yes, as |a(3) - a(4)| = 2, which is prime. We get a(4) = 1.
Can we duplicate a(2) to form a(5)? No, as |a(4) - a(5)| would be 1 and 1 is not prime. We thus extend S with the smallest integer X not yet in S such that |a(4) - X| is not prime; we get a(5) = 5.
Can we duplicate a(2) to form a(6)? Yes, as |a(6) - a(5)| = 3, which is prime; we get a(6) = 2.
Etc.
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CROSSREFS
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Cf. A306803 (obtained by replacing the absolute difference by the sum in the definition).
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KEYWORD
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AUTHOR
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STATUS
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approved
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