A173587 Primes of the form x^3 + 2y^3, with x,y >0.

3, 17, 29, 43, 127, 179, 251, 277, 359, 397, 433, 557, 593, 811, 857, 1051, 1367, 1459, 1583, 1753, 1801, 2017, 2027, 2213, 2251, 2447, 2663, 2689, 2729, 2789, 3221, 3331, 3391, 3457, 3581, 4421, 4519, 4787, 4967, 5653, 6037, 6217, 7109, 7883, 8081

Offset: 1

Author

Michel Lagneau, Feb 22 2010

Keywords

• nonn

Comments

Heath-Brown shows that this sequence is infinite.

Examples

			a(1) = 1^3+2*1^3 =3, prime. a(2) = 1^3 + 2* 2^3 = 17. a(7) = 1^3+2*r^3 =251.
		

Links

• Zak Seidov, Table of n, a(n) for n = 1..1000
• D. R. Heath-Brown, Primes represented by x^3 + 2y^3, Acta Mathematica 186 (2001), pp. 1-84.
• H. Iwaniec, Primes represented by quadratic polynomials in two variables, Acta Arithmetica, 24 (1974), 435-459
• T. Mitsui, Generalized prime number theorem, Jap.J. Math,26 (1956),1-42

Crossrefs

Cf. A007490, A219722.

Programs

Maple

T:=array(0..5000000): ind:=1: for x from 1 to 1000 do: for y from 1 to 1000 do: z:=x^3 + 2*y^3: if type(z,prime)=true then T[ind] :=z: ind :=ind+1: else fi: od: od: mini:=T[1]: ii:=1: for p from 1 to ind-1 do: for n from 1 to ind-1 do: if T[n] < mini then mini:= T[n]: ii:=n: else fi: od: print(mini): T[ii]:= 999999999999999: ii:=1: mini:=T[1] : od:

Mathematica

formQ[p_] := Reduce[0 < x < p^(1/3) && 0 < y < (p/2)^(1/3) && x^3 + 2 y^3 == p, {x, y}, Integers] =!= False; Select[ Prime[ Range[1100]], formQ] (* Jean-François Alcover, Sep 28 2011 *)

PARI

list(lim)=my(v=List(),t); for(y=1,sqrtn(lim\2,3), t=2*y^3; for(x=1,sqrtn(lim-t,3), if(isprime(t+x^3), listput(v,t+x^3)))); vecsort(Vec(v),,8) \\ Charles R Greathouse IV, Sep 28 2011

Extensions

Converted references to links - R. J. Mathar, Feb 24 2010