A109611 - OEIS

A109611

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

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; refs; listen; history; text; internal format)

	OFFSET	`1,1`
	COMMENTS	`43 is the first prime which is not a member (seeA102540).` `ContainsA001359= lesser of twin primes.` `A063637is 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 ofA037143.-Flávio V. Fernandes,May 08 2021` `Named after the Chinese mathematician Chen Jingrun (1933-1996). -Amiram Eldar,Jun 10 2021`
	LINKS	`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.`
	FORMULA	`a(n)+2 =A139690(n).` `Sum_{n>=1} 1/a(n) converges (Zhou, 2009). -Amiram Eldar,Jun 10 2021`
	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(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`
	MATHEMATICA	`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 )`
	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` `(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`
	CROSSREFS	`Union ofA001359andA063637.` `Cf.A001358,A112021,A112022,A139689,A269256.` `Cf.A037143.` `Sequence in context:A245576A086472A219669A181325A078133A341661` `Adjacent sequences:A109608A109609A109610A109612A109613A109614`
	KEYWORD	`nonn`
	AUTHOR	`Paul Muljadi,Jul 31 2005`
	EXTENSIONS	`Corrected byAlonso del Arte,Aug 08 2005`
	STATUS	`approved`