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

1,1

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

Links

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

Example

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

Mathematica

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

Prog

(PARI) g(n, k) = forprime(x1=3, n, x2=x1+2; if(isprime(x2), p=x1*x2+k; if(isprime(p), print1(x1", ") ) ) )
(Magma) [p: p in PrimesUpTo(25000)|  IsPrime(p+2) and IsPrime(p^2+2*p+2)] // Vincenzo Librandi, Jan 29 2011

Crossrefs

Cf. A051779.

Sequence in context: A225672 A355803 A240125 * A087307 A280427 A038535

Adjacent sequences: A108010 A108011 A108012 * A108014 A108015 A108016

Keyword

easy,nonn

Author

Cino Hilliard, May 30 2005

Status

approved