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

1,2

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).

Links

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

Example

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.

Mathematica

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

Prog

(PARI) for(n=1, 100000, 2; if(isprime(n+2) && isprime(n+4) && isprime(n^2+2) &&isprime(n^2+4), print1(n, ", ")))
(Magma) [n: n in [0..100000] | IsPrime(n+2) and IsPrime(n+4) and IsPrime(n^2+2) and IsPrime(n^2+4)];
(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

Crossrefs

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

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

Sequence in context: A367520 A346542 A033853 * A049187 A049161 A358685

Adjacent sequences: A284011 A284012 A284013 * A284015 A284016 A284017

Keyword

nonn

Author

K. D. Bajpai, Mar 18 2017

Status

approved