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

1,1

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.

Links

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

Example

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

Mathematica

f[p_] := p*(p + 2) + 6; f /@ Select[Range[10^4], And @@ PrimeQ[{#, # + 2, f[#]}] &] (* Amiram Eldar, Mar 26 2021 *)

Prog

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

Crossrefs

Cf. A001359, A051779.

Sequence in context: A337872 A192821 A141957 * A142630 A259835 A082252

Adjacent sequences: A108013 A108014 A108015 * A108017 A108018 A108019

Keyword

easy,nonn

Author

Cino Hilliard, May 31 2005

Status

approved