A086081 Numbers m such that m and its 2's complement are both primes. In other words, m and 2^k - m (where k is the smallest power of 2 such that 2^k > m) are primes.

2, 5, 11, 13, 19, 29, 41, 47, 53, 59, 61, 67, 97, 109, 149, 167, 173, 197, 227, 233, 239, 251, 271, 283, 313, 331, 349, 373, 409, 433, 439, 499, 509, 521, 557, 563, 593, 641, 677, 743, 761, 773, 797, 827, 857, 887, 911, 941, 953, 971, 977, 983, 1013, 1019, 1021

Offset: 1

Author

Chuck Seggelin, Jul 08 2003

Keywords

• nonn
• easy

Comments

In the first 672509 primes, 64894 of them (about 9.65%) are 2's-complement primes.

Examples

			19 is a term because 19 is prime and (2^5 - 19) = (32 - 19) = 13 which is prime.
1777 is a term because 1777 is prime and (2^11 - 1777) = (2048 - 1777) = 271 which is prime.
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A068811.

Programs

Mathematica

Join[{2}, Select[Prime[Range[250]], PrimeQ[BitXor[#, 2^Ceiling[Log[2, #]] - 1] + 1] &]] (* Alonso del Arte, Feb 12 2013 *)

PARI

select(m->isprime((2<<(log(m+.5)\log(2)))-m), primes(100)) \\ Charles R Greathouse IV, Feb 13 2013

Formula

If isPrime(p) And isPrime(2^(floor(Log(p, 2)) + 1) - p) then sequence.add(p)

If A(x) is the counting function of the terms a(n) <= x, then A(x) = O(xloglogx/(logx)^2) [From Vladimir Shevelev, Dec 04 2008]