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Lexicographically earliest sequence of distinct positive terms such that for any n > 0, a(n) XOR a(n+1) XOR a(n+2) is prime (where XOR denotes the bitwise XOR operator).
2

%I #11 Jan 07 2018 23:49:06

%S 1,2,4,3,5,11,9,7,12,6,8,13,14,16,15,18,10,19,20,22,17,24,26,21,28,30,

%T 29,38,36,31,40,32,23,42,34,25,44,48,27,41,33,35,39,43,37,51,45,49,53,

%U 47,63,57,59,55,67,61,69,77,65,75,73,81,87,79,91,71,83

%N Lexicographically earliest sequence of distinct positive terms such that for any n > 0, a(n) XOR a(n+1) XOR a(n+2) is prime (where XOR denotes the bitwise XOR operator).

%C See A297879 for the corresponding prime numbers.

%C This sequence has connections with A076990: here we combine triples of successive terms with the XOR operator, there with the usual addition operator.

%C The sequence alternates long runs of odd terms and long runs with periodic parity (even, even, odd); changes from one type of run to the other occur near terms such that a(n) XOR a(n+1) XOR a(n+2) = 2; see illustration in Links section.

%H Rémy Sigrist, <a href="/A297706/b297706.txt">Table of n, a(n) for n = 1..10000</a>

%H Rémy Sigrist, <a href="/A297706/a297706.gp.txt">PARI program for A297706</a>

%H Rémy Sigrist, <a href="/A297706/a297706.png">Colored scatterplot of the first 25000 terms</a> (where the color is function of the parity of a(n))

%e The first terms of the sequence are:

%e n a(n) a(n) XOR a(n+1) XOR a(n+2)

%e -- ---- --------------------------

%e 1 1 7

%e 2 2 5

%e 3 4 2

%e 4 3 13

%e 5 5 7

%e 6 11 5

%e 7 9 2

%e 8 7 13

%e 9 12 2

%e 10 6 3

%e 11 8 11

%e 12 13 19

%e 13 14 17

%e 14 16 13

%e 15 15 23

%e 16 18 11

%e 17 10 13

%e 18 19 17

%e 19 20 19

%e 20 22 31

%o (PARI) See Links section.

%Y Cf. A076990, A297879.

%K nonn,base

%O 1,2

%A _Rémy Sigrist_, Jan 03 2018