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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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72
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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.
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
Named after the Chinese mathematician Chen Jingrun (1933-1996). - Amiram Eldar, Jun 10 2021
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LINKS
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Eric Weisstein's World of Mathematics, Chen Prime.
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FORMULA
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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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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;
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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
(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)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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