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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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71
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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. - 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
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LINKS
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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.
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FORMULA
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a(n)+2 = A139690(n).
Sum_{n>=1} 1/a(n) converges (Zhou, 2009). - Amiram Eldar, Jun 10 2021
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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 *)
SequencePosition[PrimeOmega[Range[500]], {1, _, 1|2}][[All, 1]] (* Jean-François Alcover, Feb 10 2018 *)
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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
(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
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CROSSREFS
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Union of A001359 and A063637.
Cf. A001358, A112021, A112022, A139689, A269256.
Cf. A037143.
Sequence in context: A245576 A086472 A219669 * A181325 A078133 A341661
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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