A125821 Numbers k for which 8*k+5 and 8*k+7 are twin primes.

3, 12, 18, 24, 33, 57, 102, 132, 153, 159, 162, 234, 243, 249, 267, 279, 288, 297, 318, 348, 423, 432, 444, 447, 477, 489, 519, 528, 552, 564, 579, 627, 684, 687, 717, 774, 783, 837, 858, 918, 948, 969, 984, 993

Offset: 1

Author

Artur Jasinski, Dec 10 2006

Keywords

• nonn

Comments

From Zak Seidov, Apr 19 2008: (Start)

Proof that all numbers in this sequence are divisible by 3:

if n=(3k+1), then 8n+7=8(3k+1)+7=3(5+8 k) (composite)

if n=(3k+2), then 8n+5=8(3k+2)+5=3(7+8 k) (composite),

so if we require that both 8n+5 and 8n+7 are primes, then n=3k, hence all terms in this sequence are multiples of 3. QED. (End)

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A001109.

For a(n)/3 see A139404.

Cf. A125822, A139402, A139404.

Programs

Mathematica

Do[If[PrimeQ[8n + 5] && PrimeQ[8n + 7], Print[n]], {n, 1, 1000}]
Select[Range[3,6000,3],AllTrue[8#+{5,7},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Sep 14 2018 *)