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Least prime p such that the n+1 numbers p + 2^k - 2, k=1..n+1, are all prime.
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%I #6 Oct 26 2023 11:18:49

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

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

%C 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.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Primek-TuplesConjecture.html">k-Tuple Conjecture</a>

%e 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.

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

%Y Cf. A000918 (2^n - 2)

%K hard,nonn

%O 1,1

%A _T. D. Noe_, Sep 09 2009