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%I #12 Mar 26 2021 08:40:04
%S 41,149,5189,39209,186629,213449,1127849,1192469,1695209,2965289,
%T 3732629,4359749,4460549,5673929,6718469,7225349,11370389,12446789,
%U 12830729,14607689,14837909,16016009,17040389,17288969,20684309
%N Primes of the form p*(p+2)+6 where p and p+2 are primes.
%C Except for the first term, these numbers end in 9. p can take one of the forms 10k+1, 10k+3, 10k+7 or 10k+9. p = 10k+1 => p*(p+2)+6 = (10k+1)(10k+3)+6 = 10h+9. p can be 10k+1. p = 10k+3 => p+2 = 0 mod 5 not prime. p cannot be 10k+3. p = 10k+7 => p(p+2)+6 = (10k+7)(10k+9)+6 = 10h+9. p can be 10k+7. p = 10k+9 => p(p+2)+6 = (10k+9)*(10k+11)+6 = 0 mod 5 not prime. p cannot be 10k+9. Thus by exhaustion p(p+2)+6 ends in 9.
%H Amiram Eldar, <a href="/A108016/b108016.txt">Table of n, a(n) for n = 1..10000</a>
%e 149 = 11*13 + 6 is a term since 11, 13 and 149 are primes.
%t f[p_] := p*(p + 2) + 6; f /@ Select[Range[10^4], And @@ PrimeQ[{#, # + 2, f[#]}] &] (* _Amiram Eldar_, Mar 26 2021 *)
%o (PARI) g(n,k=6) = forprime(x1=3,n, x2=x1+2; if(isprime(x2), p=x1*x2+k; if(isprime(p), print1(p, ", ") ) ) )
%Y Cf. A001359, A051779.
%K easy,nonn
%O 1,1
%A _Cino Hilliard_, May 31 2005
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