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A306803
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An irregular fractal sequence: underline a(n) iff [a(n-1) + a(n)] is prime; all underlined terms rebuild the starting sequence.
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3
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0, 1, 3, 0, 4, 1, 5, 7, 2, 3, 0, 6, 8, 10, 11, 9, 4, 1, 13, 12, 5, 15, 17, 16, 7, 14, 18, 20, 19, 21, 2, 3, 0, 22, 23, 6, 24, 25, 26, 28, 27, 29, 8, 30, 32, 31, 10, 34, 35, 33, 36, 11, 37, 38, 9, 4, 1, 39, 41, 40, 13, 42, 43, 44, 46, 45, 47, 12, 5, 49, 50, 48, 51, 53, 52, 15, 54, 17, 55, 16, 7, 56, 58, 57, 14, 60, 59, 61, 18, 62
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OFFSET
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1,3
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COMMENTS
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The sequence S starts with a(1) = 0 and a(2) = 1. S is extended by duplicating the first term A among the not yet duplicated terms, under the condition that [A + the last term Z of the sequence] is prime. If this is not the case, we then extend the S with the smallest integer X not yet present in S such that [X + the last term Z of the sequence] is not a prime. This is the lexicographically first sequence with this property.
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LINKS
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EXAMPLE
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S starts with a(1) = 0 and a(2) = 1
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 [X + a(2)] is not prime. We get X = 3 and thus a(3) = 3.
Can we duplicate a(1) to form a(4)? Yes, as now [a(1) + a(3)] is prime; we get thus a(4) = 0.
Can we duplicate a(2) to form a(5)? No, as a(4) + a(2) would be 1 and 1 is not prime; we thus extend S with the smallest integer X not yet in S such that [X + a(4)] is not prime. We get X = 4 and thus a(5) = 4.
Can we duplicate a(2) to form a(6)? Yes, as now [a(2) + a(5)] is prime; we get thus a(6) = 1
Can we duplicate a(3) to form a(7)? No, as a(6) + a(3) would be 4 and 4 is not prime; we thus extend S with the smallest integer X not yet in S such that [X + a(6)] is not prime. We get X = 5 and thus a(7) = 5.
Can we duplicate a(3) to form a(8)? No, as a(7) + a(3) would be 8 and 8 is not prime; we thus extend S with the smallest integer X not yet in S such that [X + a(7)] is not prime. We get X = 7 and thus a(8) = 7.
Can we duplicate a(3) to form a(9)? No, as a(8) + a(3) would be 10 and 10 is not prime; we thus extend S with the smallest integer X not yet in S such that [X + a(8)] is not prime. We get X = 2 and thus a(9) = 2.
Can we duplicate a(3) to form a(10)? Yes, as now [a(3) + a(9)] is prime; we get thus a(10) = 3.
Can we duplicate a(4) to form a(11)? Yes, as [a(4) + a(10)] is prime; we get thus a(11) = 0.
Etc.
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CROSSREFS
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Cf. A306808 (which is obtained by replacing prime by palindrome in the definition).
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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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