A284014 Numbers k such that {k + 2, k + 4} and {k^2 + 2, k^2 + 4} are both twin prime pairs.

1, 3, 15, 57, 147, 2085, 6687, 6957, 11055, 15267, 17385, 17577, 20505, 20637, 23667, 26247, 31077, 31317, 32115, 32967, 34497, 39225, 47775, 52065, 53715, 55335, 56205, 58365, 62187, 63585, 66567, 67215, 70875, 77235, 77475, 82005, 85827, 89595, 89817, 107505

Offset: 1

Author

K. D. Bajpai, Mar 18 2017

Keywords

• nonn

Comments

After a(1), all the terms are multiples of 3.

After a(2), all the terms are congruent to 5 or 7 (mod 10).

Examples

			a(2) = 3, {3 + 2 = 5, 3 + 4 = 7} and {3^2 + 2 = 11, 3^2 + 4 = 13} are twin prime pairs.
a(3) = 15, {15 + 2 = 17, 15 + 4 = 19} and {15^2 + 2 = 227, 15^2 + 4 = 229} are twin prime pairs.
		

Links

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

Crossrefs

Appears to be the intersection of A086381 and A256388, but that may be unproven.

Cf. A000040, A001359, A007591, A010051, A067201, A178336, A178337.

Programs

Magma

[n: n in [0..100000] | IsPrime(n+2) and IsPrime(n+4) and IsPrime(n^2+2) and IsPrime(n^2+4)];

Mathematica

Select[Range[1000000], PrimeQ[# + 2] && PrimeQ[# + 4] && PrimeQ[#^2 + 2] && PrimeQ[#^2 + 4] &]

PARI

for(n=1, 100000,2; if(isprime(n+2) && isprime(n+4) && isprime(n^2+2) &&isprime(n^2+4), print1(n, ", ")))

Scheme

;; With Antti Karttunen's IntSeq-library.
(define A284014 (MATCHING-POS 1 1 (lambda (n) (and (= 1 (A010051 (+ n 2))) (= 1 (A010051 (+ n 4))) (= 1 (A010051 (+ (* n n) 2))) (= 1 (A010051 (+ (* n n) 4)))))))
;; Antti Karttunen, Apr 15 2017