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A108016 Primes of the form p*(p+2)+6 where p and p+2 are primes. 1
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified December 9 09:37 EST 2021. Contains 349627 sequences. (Running on oeis4.)