A108016 Primes of the form p*(p+2)+6 where p and p+2 are primes.

41, 149, 5189, 39209, 186629, 213449, 1127849, 1192469, 1695209, 2965289, 3732629, 4359749, 4460549, 5673929, 6718469, 7225349, 11370389, 12446789, 12830729, 14607689, 14837909, 16016009, 17040389, 17288969, 20684309

Offset: 1

Author

Cino Hilliard, May 31 2005

Keywords

• easy
• nonn

Comments

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.

Examples

			149 = 11*13 + 6 is a term since 11, 13 and 149 are primes.
		

Links

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

Crossrefs

Cf. A001359, A051779.

Programs

Mathematica

f[p_] := p*(p + 2) + 6; f /@ Select[Range[10^4], And @@ PrimeQ[{#, # + 2, f[#]}] &] (* Amiram Eldar, Mar 26 2021 *)
Select[Times@@#+6&/@Select[Partition[Prime[Range[1000]],2,1],#[[2]]-#[[1]]==2&],PrimeQ] (* Harvey P. Dale, Aug 16 2024 *)

PARI

g(n,k=6) = forprime(x1=3,n, x2=x1+2; if(isprime(x2), p=x1*x2+k; if(isprime(p), print1(p, ", ") ) ) )