A306581 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.

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, 16, 37, 34, 45, 22, 39, 28, 55, 46, 57, 35, 24, 43, 36, 47, 33, 32, 41, 38, 67, 30, 53, 42, 61, 40, 49, 48, 73, 50, 51, 59, 56, 69, 44, 63, 52, 77, 60, 79, 54, 65

Offset: 1

Author

Rémy Sigrist, Feb 25 2019

Keywords

• nonn
• base

Comments

This sequence is the binary variant of A228323.

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

Examples

			The first terms, alongside their binary representations, and the concatenation of consecutive terms, with prime numbers denoted by a star, are:
  n   a(n)  bin(a(n))  bin(a(n)a(n+1))  bin(a(n+1)a(n))
  --  ----  ---------  ---------------  ---------------
   1     1          1              110              101*
   2     2         10             1011*            1110
   3     3         11            11100            10011*
   4     4        100           100101*          101100
   5     5        101           101110           110101*
   6     6        110          1101011*         1011110
   7    11       1011         10111000         10001011*
   8     8       1000          1000111*         1111000
   9     7        111          1111001          1001111*
  10     9       1001         10011101*        11011001
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..10000
• Rémy Sigrist, Colored scatterplot of (n, a(n)) for n = 1..10000 (where the color corresponds to the parity of a(n): red for odd, blue for even)
• Rémy Sigrist, Colored scatterplot of (n, a(n)-n) for n = 1..1000000 (where the color corresponds to the parity of a(n)-n: red for odd, blue for even)
• Rémy Sigrist, Scatterplot of the first 1000000 terms of the ordinal transform of n -> a(n)-n
• Rémy Sigrist, PARI program for A306581

Crossrefs

See A228323 for the decimal variant.

Cf. A018800.

Programs

Mathematica

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 *)

PARI

See Links section.