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A322748 Primes p such that q=p^2+p+1 is prime and (q^2+q+1)/3 is prime. 1
2, 3, 5, 41, 71, 383, 827, 1217, 1931, 2729, 2969, 3491, 3671, 4751, 5039, 6551, 6791, 6833, 9323, 9689, 10223, 11933, 13163, 14549, 15413, 15797, 17393, 17573, 19991, 22349, 24533, 25523, 25943, 28409, 28517, 30593, 31277, 31337, 32507, 34319, 35747, 38069, 38603, 38993, 41177, 42089, 42839, 44507 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Except for 2 and 3, all terms == 5 (mod 6).  If p == 2 (mod 3), q == 1 (mod 3), and so q^2+q+1 is divisible by 3.

LINKS

Robert Israel, Table of n, a(n) for n = 1..10000

EXAMPLE

a(3)=41 is a term because 41^2+41+1=1723 is prime and (1723^2+1723+1)/3=990151 is prime.

MAPLE

filter:= proc(p) local r;

  r:= p^2 + p + 1;

isprime(p) and isprime(r) and isprime((r^2+r+1)/3)

end proc:

select(filter, [2, 3, seq(i, i=5..10^5, 6)]); # Robert Israel, Dec 25 2018

MATHEMATICA

f[n_] := n^2 + n + 1; Select[Range[45000], PrimeQ[#] && PrimeQ[f[#]] && PrimeQ[f[f[#]]/3] &] (* Amiram Eldar, Dec 25 2018 *)

PROG

(PARI) is(n) = isprime(n) && isprime(q=(n^3-1)/(n-1)) && isprime((q^3-1)/(3*q-3)) \\ David A. Corneth, Dec 25 2018

CROSSREFS

Cf. A002061.

Sequence in context: A215105 A088483 A235681 * A224781 A136015 A106713

Adjacent sequences:  A322745 A322746 A322747 * A322749 A322750 A322751

KEYWORD

nonn

AUTHOR

Robert Israel, Dec 25 2018

STATUS

approved

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Last modified October 16 18:03 EDT 2019. Contains 328102 sequences. (Running on oeis4.)