

A109611


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


64



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
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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
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


REFERENCES

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. 157176.


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], 20042005, pp. 5, 14, 1819, 21.
B. Green and T. Tao, Restriction theory of the Selberg sieve, with applications, J. ThÃ©or. 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(n1)) ;
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, 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



