A165235 Least prime p such that the n+1 numbers p + 2^k - 2, k=1..n+1, are all prime.

3, 5, 5, 17, 17, 1607, 1607, 19427, 2397347207, 153535525937

Offset: 1

Author

T. D. Noe, Sep 09 2009

Keywords

• hard
• nonn

Comments

The n+1 primes have common differences of 2^k for k=1..n. For any n, the set {2^k - 2, k=1..n+1} is admissible. Hence by the prime k-tuple conjecture, an infinite number of primes p should exist for each n. Note that a(1) is the first term of the twin primes A001359 and a(2) is the first term of prime triples A022004. The a(12) term is greater than 10^12.

Examples

			a(5)=17 because {17,19,23,31,47,79} are 6 primes whose differences are powers of 2, and 17 is the least such prime.
		

Links

• Eric Weisstein's World of Mathematics, k-Tuple Conjecture

Crossrefs

Cf. A000918 (2^n - 2)

Programs

Mathematica

p=3; Table[While[ !And@@PrimeQ[p+2^Range[2,n+1]-2], p=NextPrime[p]]; p, {n,8}]