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Search: id:A109611
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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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+0
49
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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
(list; graph; listen)
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OFFSET
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1,1
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COMMENT
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43 is the first prime which is not a member (see A102540).
Contains A001359 = lesser of twin primes.
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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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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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with(numtheory): ts_chen:= proc(n) local i, p, ans; ans:=[ ]; for i from 1 to n do if ( isprime(i) = true) then if ( isprime(i+2) = true or bigomega(i+2) = 2 ) then ans:=[ op(ans), i ] fi: fi: od: RETURN(ans); end: ts_chen(5000); - Jani Melik (jani_melik(AT)hotmail.com), May 04 2006
ts_chen:= proc(n) local i, ans; ans:=[ ]: for i from 1 to n do if ( isprime(i) = 'true') then if ( isprime(i+2) = 'true' or numtheory[bigomega](i+2) = 2) then ans:=[ op(ans), i ] fi fi od: return ans end: ts_chen(1000); - Jani Melik (jani_melik(AT)hotmail.com), May 05 2006
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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] &] (Delarte)
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PROGRAM
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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 (mathar(AT)strw.leidenuniv.nl), Aug 20 2006
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CROSSREFS
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Cf. A001358.
Cf. A112021, A112022, A139689, A139690(n)=a(n)+2.
Sequence in context: A049555 A052042 A086472 this_sequence A078133 A167773 A089189
Adjacent sequences: A109608 A109609 A109610 this_sequence A109612 A109613 A109614
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KEYWORD
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nonn
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AUTHOR
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Paul Muljadi (paulmuljadi(AT)yahoo.com), Jul 31 2005
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EXTENSIONS
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Corrected by Alonso Delarte (alonso.delarte(AT)gmail.com), Aug 08 2005
Replaced arxiv URL by non-cached version - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Oct 07 2009
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