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A109611 Chen primes: primes p such that p + 2 is either a prime or a semiprime. 58
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. [From Reinhard Zumkeller, Mar 22 2010]

REFERENCES

B. Green and T. Tao, Restriction theory of the Selberg sieve, with applications, 2005, pp. 5, 14, 18 - 19, 21

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]

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: A052042 A086472 A219669 * A181325 A078133 A238242

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 October 24 23:54 EDT 2014. Contains 248516 sequences.