A241817 Semiprimes sp such that sp-3 is prime.

6, 10, 14, 22, 26, 34, 46, 62, 74, 82, 86, 106, 134, 142, 166, 194, 202, 214, 226, 254, 274, 314, 334, 362, 382, 386, 422, 446, 466, 482, 502, 526, 566, 622, 634, 662, 694, 746, 842, 862, 866, 886, 914, 922, 974, 1042, 1094, 1126, 1154, 1174, 1226, 1234, 1262

Offset: 1

Author

K. D. Bajpai, Apr 29 2014

Keywords

• nonn

Comments

Even numbers of the form 2p, p prime, that can be expressed as the sum of two primes in at least two ways as 2p = p + p = 3 + (2p-3). For example, 34 is in the sequence because 34 = 2*17 = 17 + 17 = 3 + 31. These are the only numbers that have Goldbach partitions with both a minimum and a maximum possible difference between their prime parts, i.e., |p-p| = 0 and |(2p-3)-3| = 2p-6 respectively. - Wesley Ivan Hurt, Apr 08 2018

Examples

			a(2) = 10 = 2*5, which is semiprime and 10-3 = 7 is a prime.
a(6) = 34 = 2*17, which is semiprime and 34-3 = 31 is a prime.
		

Links

• K. D. Bajpai, Table of n, a(n) for n = 1..2500

Crossrefs

Cf. A001358, A063908, A092207, A123017, A198327.

Programs

Maple

with(numtheory): A241817:= proc(); if bigomega(x)=2 and isprime(x-3) then  RETURN (x); fi; end: seq(A241817 (), x=1..3000);

Mathematica

2 Select [Prime[Range[5!]], PrimeQ[2 # - 3] &] (* Vincenzo Librandi, Apr 10 2018 *)
Select[Range[1500],PrimeOmega[#]==2&&PrimeQ[#-3]&] (* Harvey P. Dale, Oct 14 2018 *)

Formula

a(n) = 2 * A063908(n). - Wesley Ivan Hurt, Apr 08 2018