A306803 An irregular fractal sequence: underline a(n) iff [a(n-1) + a(n)] is prime; all underlined terms rebuild the starting sequence.

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

Offset

1,3

Comments

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.

Links

Jean-Marc Falcoz, Table of n, a(n) for n = 1..10002

Example

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.

Crossrefs

Cf. A306808 (which is obtained by replacing prime by palindrome in the definition).

Sequence in context: A238573 A346965 A325491 * A319974 A138376 A077140

Adjacent sequences: A306800 A306801 A306802 * A306804 A306805 A306806

Keyword

base,nonn

Author

Alexandre Wajnberg and Eric Angelini, Mar 11 2019

Status

approved