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

Author

Carl R. White, Jun 19 2009

Keywords

• nonn

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.

Examples

			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).
		

Links

• Eric Weisstein's World of Mathematics, Prime Constellation.

Crossrefs

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