%I #16 Feb 27 2019 11:50:40

%S 1,2,3,4,5,6,11,8,7,9,13,10,17,12,25,18,23,15,14,19,20,21,26,27,31,29,

%T 16,37,34,45,22,39,28,55,46,57,35,24,43,36,47,33,32,41,38,67,30,53,42,

%U 61,40,49,48,73,50,51,59,56,69,44,63,52,77,60,79,54,65

%N Lexicographically earliest sequence of distinct positive terms such that the binary representations of two consecutive terms can always been concatenated in some order, without leading zero, to produce the binary representation of a prime number.

%C This sequence is the binary variant of A228323.

%C The sequence is well defined; the argument used to prove that A018800(n) always exists applies here also.

%H Rémy Sigrist, <a href="/A306581/b306581.txt">Table of n, a(n) for n = 1..10000</a>

%H Rémy Sigrist, <a href="/A306581/a306581_4.png">Colored scatterplot of (n, a(n)) for n = 1..10000</a> (where the color corresponds to the parity of a(n): red for odd, blue for even)

%H Rémy Sigrist, <a href="/A306581/a306581_5.png">Colored scatterplot of (n, a(n)-n) for n = 1..1000000</a> (where the color corresponds to the parity of a(n)-n: red for odd, blue for even)

%H Rémy Sigrist, <a href="/A306581/a306581_6.png">Scatterplot of the first 1000000 terms of the ordinal transform of n -> a(n)-n</a>

%H Rémy Sigrist, <a href="/A306581/a306581_1.gp.txt">PARI program for A306581</a>

%e The first terms, alongside their binary representations, and the concatenation of consecutive terms, with prime numbers denoted by a star, are:

%e n a(n) bin(a(n)) bin(a(n)a(n+1)) bin(a(n+1)a(n))

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

%e 1 1 1 110 101*

%e 2 2 10 1011* 1110

%e 3 3 11 11100 10011*

%e 4 4 100 100101* 101100

%e 5 5 101 101110 110101*

%e 6 6 110 1101011* 1011110

%e 7 11 1011 10111000 10001011*

%e 8 8 1000 1000111* 1111000

%e 9 7 111 1111001 1001111*

%e 10 9 1001 10011101* 11011001

%t a = {1}; c[x_, y_] := FromDigits[Join @@ IntegerDigits[{x, y}, 2], 2]; While[Length@a < 67, j=1; While[MemberQ[a, j] || ! (PrimeQ@ c[a[[-1]], j] || PrimeQ@ c[j, a[[-1]]]), j++]; AppendTo[a, j]]; a (* _Giovanni Resta_, Feb 27 2019 *)

%o (PARI) See Links section.

%Y See A228323 for the decimal variant.

%Y Cf. A018800.

%K nonn,base

%O 1,2

%A _Rémy Sigrist_, Feb 25 2019