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%I #17 Sep 08 2022 08:45:19
%S 3,5,149,179,239,269,419,569,1289,1319,2309,2549,2729,3359,3389,4259,
%T 4649,5849,5879,6359,6779,8999,9239,9629,10529,10889,11969,13679,
%U 13829,14009,14549,16229,16649,18059,18119,18539,19139,19379,21599,21839
%N Primes p such that p + 2 and p*(p + 2) + 2 are primes.
%C 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.
%C 10k+1 => p(p+2)+2 ends in 5, hence not prime, so p <> form 10k+1.
%C 10k+3 => (p+2) ends in 5, hence not prime, so p <> form 10k+3.
%C 10k+7 => p(p+2)+2 ends in 5, hence not prime, so p <> form 10k+7.
%C 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.
%C Subsequence of A051507. All terms larger than 5 are congruent to 29 mod 30. - _Zak Seidov_
%H Michael De Vlieger, <a href="/A108013/b108013.txt">Table of n, a(n) for n = 1..5000</a>
%e 149*151 + 2 = 22501. 149, 151, and 22501 are all prime so 149 is in the sequence.
%t Select[Prime@ Range@ 3000, AllTrue[{#2, #1 #2 + 2}, PrimeQ] & @@ {#, # + 2} &] (* _Michael De Vlieger_, Jan 22 2018 *)
%o (PARI) g(n,k) = forprime(x1=3,n, x2=x1+2; if(isprime(x2), p=x1*x2+k; if(isprime(p), print1(x1",") ) ) )
%o (Magma) [p: p in PrimesUpTo(25000)| IsPrime(p+2) and IsPrime(p^2+2*p+2)] // _Vincenzo Librandi_, Jan 29 2011
%Y Cf. A051779.
%K easy,nonn
%O 1,1
%A _Cino Hilliard_, May 30 2005
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