A176229 The smaller members p of cousin prime pairs (p,p+4) with a semiprime arithmetic mean p+2.

7, 13, 19, 37, 67, 109, 127, 307, 379, 487, 499, 769, 877, 937, 1009, 1297, 1567, 2269, 2389, 2659, 2857, 3037, 3187, 3457, 3847, 3907, 3919, 4447, 4789, 4969, 4999, 5077, 5167, 5347, 5737, 6007, 6997, 7039, 7669, 8689, 8779, 9199, 10597, 11467, 11827

Offset: 1

Author

Ulrich Krug (leuchtfeuer37(AT)gmx.de), Apr 12 2010

Keywords

• nonn

Comments

By definition a subsequence of A063637 and of A023200.

The associated p+4 are members of A063638.

Because all members of A023200 are == 1 (mod 3), the semiprimes p+2 are all == 0 (mod 3), so one of their two factors is 3.

The least-significant digit (LSD) of p > 13 in A023200 is always 3, 7 or 9, but those with LSD equal to 3 demand p+2 to have LSD 5 and therefore divisor 5 which contradicts the semiprime property above, so 13 is the only member of the sequence with LSD equal to 3.

Examples

			7 = prime(4), 11 = prime(5), (7+11)/2 = 3^2 = semiprime(3), so 7 is in the sequence.
13 = prime(6), 17 = prime(7), (13+17)/3 = 3 * 5 = semiprime(6), so 13 is in the sequence.
19 = prime(8), 23 = prime(9), (19+23)/3 = 3 * 7 = semiprime(7), so 19 is in the sequence.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A001358, A023200, A046132, A063637, A063638.

Programs

Mathematica

aQ[n_] := PrimeQ[n] && PrimeOmega[n + 2] == 2 && PrimeQ[n + 4]; Select[Range[12000], aQ] (* Amiram Eldar, Sep 08 2019 *)
Select[Prime[Range[1500]],PrimeQ[#+4]&&PrimeOmega[#+2]==2&] (* Harvey P. Dale, May 15 2023 *)