A219722 -id:A219722 - cp's OEIS frontend

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

Michel Lagneau, Feb 22 2010

nonn

Heath-Brown shows that this sequence is infinite.

			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.

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

Cf. A007490, A219722.

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:

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 *)

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

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

A219725 Integers expressible as x^3 + 2*y^3 (x, y > 0) in two ways.

Original entry on oeis.org

1459, 3391, 6758, 6875, 6913, 8317, 11672, 17826, 18523, 19034, 27128, 29845, 32581, 39393, 54064, 54125, 55000, 55304, 66536, 78733, 79939, 82522, 91557, 93376, 95519, 100171, 104073, 125054, 140610, 142608, 148184, 152272, 167399, 172565, 182375, 182466

Offset: 1

Zak Seidov, Nov 26 2012

nonn

			1459 = 1^3 + 2*9^3 = 11^3 + 2*4^3.

Cf. A173587, A219722.

A219726 Integers of the form x^3 + 2y^3 (x, y > 0).

Original entry on oeis.org

3, 10, 17, 24, 29, 43, 55, 62, 66, 80, 81, 118, 127, 129, 136, 141, 155, 179, 192, 218, 232, 251, 253, 258, 270, 277, 314, 344, 345, 359, 375, 397, 433, 440, 459, 466, 471, 496, 514, 528, 557, 566, 593, 640, 648, 687, 694, 713, 731, 745, 750, 762, 775, 783

Offset: 1

Zak Seidov, Nov 26 2012

nonn

D. R. Heath-Brown proved in 2001 that there are infinitely many prime numbers in this sequence. These primes are in A173587. - Bernard Schott, Apr 07 2020

Charles R Greathouse IV, 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.
Wikipedia, Roger Heath-Brown

Cf. A173587, A219722, A219725.

m = 10^3; Union[Flatten@Table[x^3 + 2 y^3, {x, m^(1/3)}, {y, ((m - x^3)/2)^(1/3)}]]

is(n)=for(y=1,sqrtnint((n-1)\2,3), if(ispower(n-2*y^3,3),return(1)));0 \\ Charles R Greathouse IV, Apr 07 2020

list(lim)=my(v=List(),Y); lim\=1; for(y=1,sqrtnint((lim-1)\2,3), Y=2*y^3; for(x=1,sqrtnint(lim-Y,3), listput(v,x^3+Y))); Set(v) \\ Charles R Greathouse IV, Apr 07 2020

A219728 Squares of the form x^3 + 2*y^3, with x, y > 0.

Original entry on oeis.org

81, 5041, 5184, 10000, 46225, 59049, 77841, 83521, 322624, 331776, 640000, 685584, 707281, 1265625, 2958400, 3157729, 3418801, 3674889, 3779136, 4157521, 4981824, 5345344, 5602689, 5736025, 7290000, 9150625, 9529569, 10725625, 12257001, 14776336, 15904144

Offset: 1

Zak Seidov, Nov 26 2012

nonn

Subsequence of A219726.

			81: (x = 3, y = 3), 5041: (x = 17,  y = 4).

Donovan Johnson, Table of n, a(n) for n = 1..1000

Cf. A173587, A219722, A219725, A219726.

With[{nn=300},Select[Flatten[Table[x^3+2y^3,{x,nn},{y,nn}]],IntegerQ[ Sqrt[#]]&]//Union] (* Harvey P. Dale, Oct 24 2018 *)

A219783 Smallest integer expressible as x^3 + 2*y^3 (x, y > 0) in exactly n ways.

Original entry on oeis.org

3, 1459, 314953199, 575012796875

Offset: 1

Zak Seidov, Nov 27 2012

a(5) > 10^17. - Donovan Johnson, Nov 29 2012

			3: (x, y) = (1, 1)
1459: (x,y) = (1,9), (11,4)
314953199: (x,y) = (121, 539), (463, 476), (649, 275)
575012796875: (x, y) = (275, 6600), (6413, 5379), (7225, 4625), (8195, 2310)

Cf. A219722, A219725, A219726, A219728.

A219784 Integers expressible as x^3 + 2*y^3 (x, y > 0) exactly in three ways.

Original entry on oeis.org

314953199, 432015625, 924667442, 1664304379, 1953968750, 2519625592, 3456125000, 4600102375, 7397339536, 8503736373, 11664421875, 13314435032, 15631750000, 20157004736, 24966020934, 27649000000, 36800819000, 38000696086, 39369149875, 44936218233, 52757156250, 54001953125, 59178716288

Offset: 1

Zak Seidov, Nov 27 2012

nonn

Smallest integer with 4 ways: 575012796875: {x, y} = {8195, 2310}, {7225, 4625}, {6413, 5379}, {275, 6600}.

			314953199: (x,y) = (121, 539), (463, 476), (649, 275).

Cf. A219722, A219725, A219726, A219728, A219783.

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Extensions

A219725 Integers expressible as x^3 + 2*y^3 (x, y > 0) in two ways.

Views

Author

Keywords

Examples

Crossrefs

A219726 Integers of the form x^3 + 2y^3 (x, y > 0).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

PARI

A219728 Squares of the form x^3 + 2*y^3, with x, y > 0.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A219783 Smallest integer expressible as x^3 + 2*y^3 (x, y > 0) in exactly n ways.

Views

Author

Keywords

Comments

Examples

Crossrefs

A219784 Integers expressible as x^3 + 2*y^3 (x, y > 0) exactly in three ways.

Views

Author

Keywords

Comments

Examples

Crossrefs