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A109611
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Chen primes: primes p such that p + 2 is either a prime or a semiprime.
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57
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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
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1,1
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COMMENTS
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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]
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REFERENCES
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B. Green and T. Tao, Restriction theory of the Selberg sieve, with applications, 2005, pp. 5, 14, 18 - 19, 21
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LINKS
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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
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FORMULA
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a(n)+2 = A139690(n).
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EXAMPLE
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a(4) = 7 because 7 + 2 = 9 and 9 is a semiprime.
a(5) = 11 because 11 + 2 = 13, a prime.
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MAPLE
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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
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MATHEMATICA
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semiPrimeQ[x_] := TrueQ[Plus @@ Last /@ FactorInteger[ x ] == 2]; Select[Prime[Range[100]], PrimeQ[ # + 2] || semiPrimeQ[ # + 2] &] (* Alonso del Arte, Aug 08 2005 *)
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PROG
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(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*/
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CROSSREFS
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Cf. A001358.
Cf. A112021, A112022, A139689.
Sequence in context: A052042 A086472 A219669 * A181325 A078133 A197298
Adjacent sequences: A109608 A109609 A109610 * A109612 A109613 A109614
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KEYWORD
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nonn
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AUTHOR
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Paul Muljadi, Jul 31 2005
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EXTENSIONS
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Corrected by Alonso del Arte, Aug 08 2005
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STATUS
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approved
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