A171135 Smallest prime p such that p + 2n is prime or semiprime.

2, 2, 3, 2, 3, 2, 3, 3, 3, 2, 3, 2, 3, 3, 3, 2, 3, 2, 3, 3, 5, 2, 3, 3, 3, 3, 3, 2, 3, 2, 3, 3, 3, 3, 3, 2, 3, 3, 5, 2, 3, 2, 3, 3, 3, 2, 3, 5, 3, 3, 5, 2, 3, 3, 3, 3, 5, 2, 3, 2, 5, 3, 3, 3, 3, 2, 3, 3, 3, 2, 3, 2, 3, 3, 5, 3, 3, 2, 3, 3, 5, 2, 3, 5, 3, 5, 3, 2, 3, 3, 3, 3, 5, 3, 3, 2, 3, 3, 3, 2, 3, 2, 3, 3, 3

Offset

1,1

Comments

For any even integer h, there exist infinitely many primes p such that p+h is either a prime or a semiprime;

A171136(n) = a(n) + 2*n;

a(A171137(n))=A000040(n) and a(m)<>A000040(n) for m < A171137(n).

Links

R. Zumkeller, Table of n, a(n) for n = 1..10000

Eric Weisstein's World of Mathematics, Chen Prime

Wikipedia, Chen prime

Formula

A010051(a(n)) * (A010051(a(n)+2*n) + A064911(a(n)+2*n)) = 1.

Mathematica

spp[n_]:=Module[{p=2}, While[PrimeOmega[p+2n]>2, p=NextPrime[p]]; p]; Array[ spp, 120] (* Harvey P. Dale, Sep 25 2018 *)

Prog

(PARI) a(n)=forprime(p=2, , if(bigomega(p+2*n)<3, return(p))) \\ Charles R Greathouse IV, Oct 21 2014
(Haskell)
a171135 n = head [p | p <- a000040_list, let x = p + 2 * n,
                      a064911 x == 1 || a010051' x == 1]
-- Reinhard Zumkeller, Oct 21 2014

Crossrefs

Cf. A171136, A171137.

Cf. A000040, A010051, A064911.

Sequence in context: A319396 A049237 A162361 * A073855 A306312 A350065

Adjacent sequences: A171132 A171133 A171134 * A171136 A171137 A171138

Keyword

nonn,changed

Author

Reinhard Zumkeller, Dec 04 2009

Status

approved