A063637 Primes p such that p+2 is a semiprime.

2, 7, 13, 19, 23, 31, 37, 47, 53, 67, 83, 89, 109, 113, 127, 131, 139, 157, 167, 181, 199, 211, 233, 251, 257, 263, 293, 307, 317, 337, 353, 359, 379, 389, 401, 409, 443, 449, 467, 479, 487, 491, 499, 503, 509, 541, 557, 563, 571, 577, 587, 631, 647, 653, 677

Offset

1,1

Comments

Primes of the form p*q - 2, where p and q are primes.

Union of A049002 and A115093. - T. D. Noe, Mar 01 2006

References

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.

Links

K. D. Bajpai, Table of n, a(n) for n = 1..14190 (first 1000 terms from T. D. Noe)

P. Pollack, Analytic and Combinatorial Number Theory Course Notes, p. 146. [?Broken link]

P. Pollack, Analytic and Combinatorial Number Theory Course Notes, p. 146.

T. Tao, Obstructions to uniformity and arithmetic patterns in the primes, arXiv:math/0505402 [math.NT], 2005.

Formula

a(n) = A062721(n) - 2.

A010051(a(n)) * A064911(a(n) + 2) = 1. - Reinhard Zumkeller, Nov 15 2011

Example

From K. D. Bajpai, Sep 06 2014: (Start)
a(3) = 13, which is prime, and 13 + 2 = 15 = 3 * 5, which is a semiprime.
a(4) = 19, which is prime, and 19 + 2 = 21 = 3 * 7, which is a semiprime.
(End)

Maple

select(t -> isprime(t) and numtheory:-bigomega(t+2)=2, [2, seq(2*i+1, i=1..500)]); # Robert Israel, Sep 07 2014

Mathematica

f[n_] := Plus @@ Flatten[ Table[ # [[2]], {1}] & /@ FactorInteger[ n]]; Select[ Prime[ Range[ 123]], f[ # + 2] == 2 &] (* Robert G. Wilson v, Apr 30 2005 *)
Select[Prime[Range[500]], PrimeOmega[#+2]==2&]  (* K. D. Bajpai, Sep 06 2014 *)

Prog

(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
(Haskell)
a063637 n = a063637_list !!(n-1)
a063637_list = filter ((== 1) . a064911 . (+ 2)) a000040_list
-- Reinhard Zumkeller, Nov 15 2011

Crossrefs

Cf. A005383, A001358, A063638.

Cf. A109611 (Chen primes).

Sequence in context: A007821 A156007 A067774 * A216526 A020623 A109346

Adjacent sequences: A063634 A063635 A063636 * A063638 A063639 A063640

Keyword

nonn

Author

Reinhard Zumkeller, Jul 21 2001

Status

approved