A161781 Binary encodings of prime constellations

1, 3, 5, 9, 11, 13, 17, 19, 25, 27, 33, 37, 41, 45, 65, 67, 69, 73, 75, 77, 81, 83, 89, 91, 97, 101, 105, 109, 129, 131, 137, 139, 145, 147, 153, 193, 195, 201, 203, 209, 211, 257, 261, 265, 269, 289, 293, 297, 301, 321, 325, 329, 333, 353, 357, 361, 365, 513, 515

Offset

0,2

Comments

Each constellation is encoded by means of dividing each of the increments to p in the k-tuple by two, raising two to the power of each and then summing the result. e.g.:

. (p,p+2,p+6) -> p+(0,2,6) => (0,1,3) -> 2^0 + 2^1 + 2^3 = 11

Each encoding is unique and so can be reversed e.g.:

. 89 = 2^0 + 2^3 + 2^4 + 2^6 -> (0,3,4,6) => (p,p+6,p+8,p+12)

Those constellations that represent all moduli for all their matching primes p are not counted; For example, encoding #7, which implies (p,p+2,p+4) only matches the prime triple (3,5,7) which is (0,2,1) mod 3, and so is not a valid constellation, and thus 7 is not in the list. Encoding #155 is the first that fails modulo 5, and is also not in the list.

Links

Table of n, a(n) for n=0..58.

Eric Weisstein's World of Mathematics, Prime Constellation

Example

Encoding #1 corresponds to the primes themselves (constellations of one), #3 corresponds to the twin primes (p,p+2), #5 to the cousin primes (p,p+4) and #9 to the 'sexy' primes (p,p+6).

Crossrefs

Cf. A008407, A020497, A094660, A135311. Also compare A014657 which is unrelated but remarkably similar

Sequence in context: A287344 A047623 A190523 * A007950 A034936 A204657

Adjacent sequences: A161778 A161779 A161780 * A161782 A161783 A161784

Keyword

nonn

Author

Carl R. White, Jun 19 2009

Status

approved