A236574 Primes p with prime(p)^3 + 2*p^3 and p^3 + 2*prime(p)^3 both prime.

3, 79, 997, 2657, 3697, 4513, 6947, 8887, 9547, 16187, 22697, 26479, 31319, 37463, 39139, 39887, 43573, 43987, 45667, 47387, 47743, 47819, 48221, 54217, 56923, 57373, 74017, 74149, 74707, 75533, 93251, 100043

Offset

1,1

Comments

Conjecture: This sequence has infinitely many terms.

In 2001 Heath-Brown proved that there are infinitely many primes of the form x^3 + 2*y^3 with x and y positive integers.

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

D. R. Heath-Brown, Primes represented by x^3 + 2y^3. Acta Mathematica 186 (2001), pp. 1-84.

Example

a(1) = 3 since prime(3)^3 + 2*3^3 = 125 + 54 = 179 and 3^3 + 2*prime(3)^3 = 27 + 2*125 = 277 are both prime, but 2^3 + 2*prime(2)^3 = 62 is composite.

Mathematica

p[n_]:=PrimeQ[Prime[n]^3+2*n^3]&&PrimeQ[n^3+2*Prime[n]^3]
n=0; Do[If[p[Prime[k]], n=n+1; Print[n, " ", Prime[k]]], {k, 1, 10000}]
Select[Prime[Range[10000]], AllTrue[{Prime[#]^3+2*#^3, #^3+2*Prime[ #]^3}, PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Aug 20 2017 *)

Crossrefs

Cf. A000040, A000578, A173587, A220413, A236193.

Sequence in context: A064456 A347165 A073176 * A367249 A062660 A331673

Adjacent sequences: A236571 A236572 A236573 * A236575 A236576 A236577

Keyword

nonn

Author

Zhi-Wei Sun, Jan 29 2014

Status

approved