A109611 Chen primes: primes p such that p + 2 is either a prime or a semiprime.

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

Offset: 1

Author

Paul Muljadi, Jul 31 2005

Keywords

• nonn

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

In 1966 Chen proved that this sequence is infinite; his proof did not appear until 1973 due to the Cultural Revolution. - Charles R Greathouse IV, Jul 12 2016

Primes p such that p + 2 is a term of A037143. - Flávio V. Fernandes, May 08 2021

Named after the Chinese mathematician Chen Jingrun (1933-1996). - Amiram Eldar, Jun 10 2021

Examples

			a(4) = 7 because 7 + 2 = 9 and 9 is a semiprime.
a(5) = 11 because 11 + 2 = 13, a prime.
		

Links

• R. J. Mathar, Table of n, a(n) for n = 1..34076
• Jing Run Chen, On the representation of a larger even integer as the sum of a prime and the product of at most two primes, Sci. Sinica 16 (1973), pp. 157-176.
• Ben Green and Terence Tao, Restriction theory of the Selberg sieve, with applications, arXiv:math/0405581 [math.NT], 2004-2005, pp. 5, 14, 18-19, 21.
• Ben Green and Terence Tao, Restriction theory of the Selberg sieve, with applications, J. Théor. Nombres Bordeaux, Vol. 18, No. 1 (2006), pp. 147-182.
• Eric Weisstein's World of Mathematics, Chen's Theorem.
• Eric Weisstein's World of Mathematics, Chen Prime.
• Wikipedia, Chen prime.
• Binbin Zhou, The Chen primes contain arbitrarily long arithmetic progressions, Acta Arithmetica, Vol. 138 (2009), pp. 301-315.

Crossrefs

Union of A001359 and A063637.

Cf. A001358, A112021, A112022, A139689, A269256.

Cf. A037143.

Programs

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 *)
SequencePosition[PrimeOmega[Range[500]], {1, , 1|2}][[All, 1]] (* _Jean-François Alcover, Feb 10 2018 *)

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

PARI

list(lim)=my(v=List([2]),semi=List(),L=lim+2,p=3); forprime(q=3,L\3, forprime(r=3,min(L\q,q), listput(semi,q*r))); semi=Set(semi); forprime(q=7,lim, if(setsearch(semi,q+2), listput(v,q))); forprime(q=5,L, if(q-p==2, listput(v,p)); p=q); Set(v) \\ Charles R Greathouse IV, Aug 25 2017

Python

from sympy import isprime, primeomega
def ok(n): return isprime(n) and (primeomega(n+2) < 3)
print(list(filter(ok, range(1, 410)))) # Michael S. Branicky, May 08 2021

Formula

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

Sum_{n>=1} 1/a(n) converges (Zhou, 2009). - Amiram Eldar, Jun 10 2021

Extensions

Corrected by Alonso del Arte, Aug 08 2005