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A109611 Chen primes: primes p such that p + 2 is either a prime or a semiprime. 63
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 47, 53, 59, 67, 71, 83, 89, 101, 107, 109, 113, 127, 131, 137, 139, 149, 157, 167, 179, 181, 191, 197, 199, 211, 227, 233, 239, 251, 257, 263, 269, 281, 293, 307, 311, 317, 337, 347, 353, 359, 379, 389, 401, 409 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

43 is the first prime which is not a member (see A102540).

Contains A001359 = lesser of twin primes.

A063637 is a subsequence. - Reinhard Zumkeller, Mar 22 2010

LINKS

R. J. Mathar, Table of n, a(n) for n = 1..34076

B. Green and T. Tao, Restriction theory of the Selberg sieve, with applications, arXiv:math/0405581 [math.NT], 2004-2005, pp. 5, 14, 18-19, 21.

B. Green and T. Tao, Restriction theory of the Selberg sieve, with applications, J. Théor. Nombres Bordeaux 18, 2006.

Eric Weisstein's World of Mathematics, Chen's Theorem

Eric Weisstein's World of Mathematics, Chen Prime

Wikipedia, Chen prime

FORMULA

a(n)+2 = A139690(n).

EXAMPLE

a(4) = 7 because 7 + 2 = 9 and 9 is a semiprime.

a(5) = 11 because 11 + 2 = 13, a prime.

MAPLE

A109611 := proc(n)

    option remember;

    if n =1 then

        2;

    else

        a := nextprime(procname(n-1)) ;

        while true do

            if isprime(a+2) or numtheory[bigomega](a+2) = 2 then

                return a;

            end if;

            a := nextprime(a) ;

        end do:

    end if;

end proc: # R. J. Mathar, Apr 26 2013

MATHEMATICA

semiPrimeQ[x_] := TrueQ[Plus @@ Last /@ FactorInteger[ x ] == 2]; Select[Prime[Range[100]], PrimeQ[ # + 2] || semiPrimeQ[ # + 2] &] (* Alonso del Arte, Aug 08 2005 *)

PROG

(PARI) isA001358(n)={ if( bigomega(n)==2, return(1), return(0) ); } isA109611(n)={ if( ! isprime(n), return(0), if( isprime(n+2), return(1), return( isA001358(n+2)) ); ); } { n=1; for(i=1, 90000, p=prime(i); if( isA109611(p), print(n, " ", p); n++; ); ); } /* R. J. Mathar, Aug 20 2006 */

CROSSREFS

Cf. A001358.

Cf. A112021, A112022, A139689.

Sequence in context: A245576 A086472 A219669 * A181325 A078133 A268513

Adjacent sequences:  A109608 A109609 A109610 * A109612 A109613 A109614

KEYWORD

nonn

AUTHOR

Paul Muljadi, Jul 31 2005

EXTENSIONS

Corrected by Alonso del Arte, Aug 08 2005

STATUS

approved

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Last modified May 1 18:06 EDT 2016. Contains 272287 sequences.