A063638 Primes p such that p-2 is a semiprime.

11, 17, 23, 37, 41, 53, 59, 67, 71, 79, 89, 97, 113, 131, 157, 163, 179, 211, 223, 239, 251, 269, 293, 307, 311, 331, 337, 367, 373, 379, 383, 397, 409, 419, 439, 449, 487, 491, 499, 503, 521, 547, 593, 599, 613, 631, 673, 683, 691, 701, 709, 719, 733, 739

Offset: 1

Author

Reinhard Zumkeller, Jul 21 2001

Keywords

• nonn

Comments

Primes of form p*q + 2, where p and q are primes.

11 is the only prime of this form where p=q. For prime p>3, 3 divides p^2+2. - T. D. Noe, Mar 01 2006

The asymptotic growth of this sequence is relevant for A204142. We have a(10^k) = (11, 79, 1571, 27961, 407741, 5647823, ...). - M. F. Hasler, Feb 13 2012

Links

• M. F. Hasler, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A005385, A001358, A063637, A109611 (Chen primes), A204142, A064911, A040976, A241809.

Programs

Haskell

a063638 n = a063638_list !! (n-1)
a063638_list = map (+ 2) $ filter ((== 1) . a064911) a040976_list
-- Reinhard Zumkeller, Feb 22 2012

Mathematica

Take[Select[ # + 2 & /@ Union[Flatten[Outer[Times, Prime[Range[100]], Prime[Range[100]]]]], PrimeQ], 60]
Select[Prime[Range[200]],PrimeOmega[#-2]==2&] (* Paolo Xausa, Oct 30 2023 *)

PARI

n=0; for (m=2, 10^9, p=prime(m); if (bigomega(p - 2) == 2, write("b063638.txt", n++, " ", p); if (n==1000, break))) \\ Harry J. Smith, Aug 26 2009

PARI

forprime(p=3,9999, bigomega(p-2)==2 & print1(p","))

PARI

p=2; for(n=1,1e4, until(bigomega(-2+p=nextprime(p+1))==2,); write("b063638.txt", n" "p)) \\ M. F. Hasler, Feb 13 2012

PARI

list(lim)=my(v=List(), t); forprime(p=3, (lim-2)\3, forprime(q=3, min((lim-2)\p, p), t=p*q+2; if(isprime(t), listput(v, t)))); Set(v) \\ Charles R Greathouse IV, Aug 05 2016

Formula

a(n) = A241809(n) + 2. - Hugo Pfoertner, Oct 30 2023