%I
%S 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,
%T 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,
%U 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
%N An irregular fractal sequence: underline a(n) iff the absolute difference a(n1)  a(n) is prime; all underlined terms rebuild the starting sequence.
%C 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(n1)  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(n1)  a(n) is not prime. S is the lexicographically earliest sequence with this property.
%H JeanMarc Falcoz, <a href="/A306807/b306807.txt">Table of n, a(n) for n = 1..10002</a>
%e S starts with a(1) = 1 and a(2) = 2
%e 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.
%e Can we duplicate a(1) to form a(4)? Yes, as a(3)  a(4) = 2, which is prime. We get a(4) = 1.
%e 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.
%e Can we duplicate a(2) to form a(6)? Yes, as a(6)  a(5) = 3, which is prime; we get a(6) = 2.
%e Etc.
%Y Cf. A306803 (obtained by replacing the absolute difference by the sum in the definition).
%K base,nonn,look
%O 1,2
%A _Alexandre Wajnberg_ and _Eric Angelini_, Mar 11 2019
