A108013 Primes p such that p + 2 and p*(p + 2) + 2 are primes.

3, 5, 149, 179, 239, 269, 419, 569, 1289, 1319, 2309, 2549, 2729, 3359, 3389, 4259, 4649, 5849, 5879, 6359, 6779, 8999, 9239, 9629, 10529, 10889, 11969, 13679, 13829, 14009, 14549, 16229, 16649, 18059, 18119, 18539, 19139, 19379, 21599, 21839

Offset: 1

Author

Cino Hilliard, May 30 2005

Keywords

• easy
• nonn

Comments

Except for the first 2 terms, these numbers all end in 9. Proof: Any odd prime p>5 can have one of the following forms: 10k+1, 10k+3, 10k+7, 10k+9.

10k+1 => p(p+2)+2 ends in 5, hence not prime, so p <> form 10k+1.

10k+3 => (p+2) ends in 5, hence not prime, so p <> form 10k+3.

10k+7 => p(p+2)+2 ends in 5, hence not prime, so p <> form 10k+7.

Thus p is of the form 10k+9 as stated. Moreover, p+2 ends in 1 and p(p+2)+2 is of the form 100h+1 since (10k+9)(10k+11)+2 = 100(k^2+2k+1)+1.

Subsequence of A051507. All terms larger than 5 are congruent to 29 mod 30. - Zak Seidov

Examples

			149*151 + 2 = 22501. 149, 151, and 22501 are all prime so 149 is in the sequence.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..5000

Crossrefs

Cf. A051779.

Programs

Magma

[p: p in PrimesUpTo(25000)|  IsPrime(p+2) and IsPrime(p^2+2*p+2)] // Vincenzo Librandi, Jan 29 2011

Mathematica

Select[Prime@ Range@ 3000, AllTrue[{#2, #1 #2 + 2}, PrimeQ] & @@ {#, # + 2} &] (* Michael De Vlieger, Jan 22 2018 *)

PARI

g(n,k) = forprime(x1=3,n, x2=x1+2; if(isprime(x2), p=x1*x2+k; if(isprime(p), print1(x1",") ) ) )