A159843 Sums of two rational cubes.

1, 2, 6, 7, 8, 9, 12, 13, 15, 16, 17, 19, 20, 22, 26, 27, 28, 30, 31, 33, 34, 35, 37, 42, 43, 48, 49, 50, 51, 53, 54, 56, 58, 61, 62, 63, 64, 65, 67, 68, 69, 70, 71, 72, 75, 78, 79, 84, 85, 86, 87, 89, 90, 91, 92, 94, 96, 97, 98, 103, 104, 105, 106, 107, 110, 114, 115, 117

Offset: 1

Author

Steven Finch, Apr 23 2009

Keywords

• nice
• nonn

Comments

Conjectured asymptotic (based on the random matrix theory) is given in Cohen (2007) on p. 378.

The prime elements are listed in A166246. - Max Alekseyev, Oct 10 2009

Alpöge et al. prove 'that the density of integers expressible as the sum of two rational cubes is strictly positive and strictly less than 1.' The authors remark that it is natural to conjecture that these integers 'have natural density exactly 1/2.' - Peter Luschny, Nov 30 2022

Jha, Majumdar, & Sury prove that every nonzero residue class mod p (for prime p) has infinitely many elements, as do 1 and 8 mod 9. - Charles R Greathouse IV, Jan 24 2023

Alpöge, Bhargava, & Shnidman prove that the lower density of this sequence is at least 2/21 and its upper density is at most 5/6. - Charles R Greathouse IV, Feb 15 2023

References

• H. Cohen, Number Theory. I, Tools and Diophantine Equations, Springer-Verlag, 2007, p. 379.

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
• Levent Alpöge, Manjul Bhargava, and Ari Shnidman, Integers expressible as the sum of two rational cubes, arXiv:2210.10730 [math.NT], Oct. 2022.
• Bogdan Grechuk, Existence of Non-Trivial Solutions to Homogeneous Equations, Polynomial Diophantine Equations, Springer, Cham (2024), Chapter 6, 473-536.
• Somnath Jha, Dipramit Majumdar, and B. Sury, Infinitely many primes in each of the residue classes 1 and 8 modulo 9 are sums of two rational cubes, arXiv preprint arXiv:2301.06970 [math.NT], 2023-2024.
• Index entries for sequences related to sums of cubes

Crossrefs

Complement of A185345.

Subsequences include A045980, A004999, and A003325.

Cf. A020894, A020895, A020897, A020898.

Programs

Mathematica

(* A naive program with a few pre-computed terms *) nmax = 117; xmax = 2000; CubeFreePart[n_] := Times @@ Power @@@ ({#[[1]], Mod[#[[2]], 3]} & /@ FactorInteger[n]); nn = Join[{1}, Reap[ Do[n = CubeFreePart[x*y*(x + y)]; If[1 < n <= nmax, Sow[n]], {x, 1, xmax}, {y, x, xmax}]][[2, 1]] // Union]; A159843 = Select[ Union[nn, nn*2^3, nn*3^3, nn*4^3, {17, 31, 53, 67, 71, 79, 89, 94, 97, 103, 107}], # <= nmax &] (* Jean-François Alcover, Apr 03 2012 *)

PARI

is(n, f=factor(n))=my(c=prod(i=1, #f~, f[i, 1]^(f[i, 2]\3)), r=n/c^3, E=ellinit([0, 16*r^2]), eri=ellrankinit(E), mwr=ellrank(eri), ar); if(r<3 || mwr[1], return(1)); if(mwr[2]<1, return(0)); ar=ellanalyticrank(E)[1]; if(ar<2, return(ar)); for(effort=1,99, mwr=ellrank(eri,effort); if(mwr[1]>0, return(1), mwr[2]<1, return(0))); "yes under BSD conjecture" \\ Charles R Greathouse IV, Dec 02 2022

Formula

A cubefree integer c>2 is in this sequence iff the elliptic curve y^2=x^3+16*c^2 has positive rank. - Max Alekseyev, Oct 10 2009