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%I #44 Nov 02 2023 12:32:41
%S 2,7,13,19,23,31,37,47,53,67,83,89,109,113,127,131,139,157,167,181,
%T 199,211,233,251,257,263,293,307,317,337,353,359,379,389,401,409,443,
%U 449,467,479,487,491,499,503,509,541,557,563,571,577,587,631,647,653,677
%N Primes p such that p+2 is a semiprime.
%C Primes of the form p*q - 2, where p and q are primes.
%C Union of A049002 and A115093. - _T. D. Noe_, Mar 01 2006
%D J.-R. Chen, On the representation of a large even integer as the sum of a prime and a product of at most two primes, Sci. Sinica 16 (1973), 157-176.
%H K. D. Bajpai, <a href="/A063637/b063637.txt">Table of n, a(n) for n = 1..14190</a> (first 1000 terms from T. D. Noe)
%H P. Pollack, <a href="http://www.math.dartmouth.edu/~ppollack/notes.pdf">Analytic and Combinatorial Number Theory</a> Course Notes, p. 146. [?Broken link]
%H P. Pollack, <a href="http://alpha01.dm.unito.it/personalpages/cerruti/ac/notes.pdf">Analytic and Combinatorial Number Theory</a> Course Notes, p. 146.
%H T. Tao, <a href="http://arXiv.org/abs/math.NT/0505402">Obstructions to uniformity and arithmetic patterns in the primes</a>, arXiv:math/0505402 [math.NT], 2005.
%F a(n) = A062721(n) - 2.
%F A010051(a(n)) * A064911(a(n) + 2) = 1. - _Reinhard Zumkeller_, Nov 15 2011
%e From _K. D. Bajpai_, Sep 06 2014: (Start)
%e a(3) = 13, which is prime, and 13 + 2 = 15 = 3 * 5, which is a semiprime.
%e a(4) = 19, which is prime, and 19 + 2 = 21 = 3 * 7, which is a semiprime.
%e (End)
%p select(t -> isprime(t) and numtheory:-bigomega(t+2)=2, [2, seq(2*i+1,i=1..500)]); # _Robert Israel_, Sep 07 2014
%t f[n_] := Plus @@ Flatten[ Table[ # [[2]], {1}] & /@ FactorInteger[ n]]; Select[ Prime[ Range[ 123]], f[ # + 2] == 2 &] (* _Robert G. Wilson v_, Apr 30 2005 *)
%t Select[Prime[Range[500]],PrimeOmega[#+2]==2&] (* _K. D. Bajpai_, Sep 06 2014 *)
%o (PARI) { n=0; for (m=1, 10^9, p=prime(m); if (bigomega(p + 2) == 2, write("b063637.txt", n++, " ", p); if (n==1000, break)) ) } \\ _Harry J. Smith_, Aug 26 2009
%o (Haskell)
%o a063637 n = a063637_list !!(n-1)
%o a063637_list = filter ((== 1) . a064911 . (+ 2)) a000040_list
%o -- _Reinhard Zumkeller_, Nov 15 2011
%Y Cf. A005383, A001358, A063638.
%Y Cf. A109611 (Chen primes).
%K nonn
%O 1,1
%A _Reinhard Zumkeller_, Jul 21 2001
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