A045980 Numbers of the form x^3 + y^3 or x^3 - y^3.

0, 1, 2, 7, 8, 9, 16, 19, 26, 27, 28, 35, 37, 54, 56, 61, 63, 64, 65, 72, 91, 98, 117, 124, 125, 126, 127, 128, 133, 152, 169, 189, 208, 215, 216, 217, 218, 224, 243, 250, 271, 279, 280, 296, 316, 331, 335, 341, 342, 343, 344, 351, 370, 386, 387, 397, 407, 432

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

Comments

Sums of two integer cubes. - Charles R Greathouse IV, Mar 30 2022

Examples

			7 = (2)^3 + (-1)^3.
		

References

• B. C. Berndt, Ramanujan's Notebooks Part IV, Springer-Verlag, see p. 86.

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000
• Chris Bispels, Matthew Cohen, Joshua Harrington, Joshua Lowrance, Kaelyn Pontes, Leif Schaumann, and Tony W. H. Wong, A further investigation on covering systems with odd moduli, arXiv:2507.16135 [math.NT], 2025. See p. 3.
• Kevin A. Broughan, Characterizing the sum of two cubes, J. Integer Seqs., Vol. 6, 2003.
• M. Kim, Diophantine equations in two variables, arXiv:math/0210329 [math.NT], 2002.
• Index entries for sequences related to sums of cubes

Crossrefs

A004999 and A003325 are subsequences.

Cf. A222304, A222305, A222306, A027750, A010052.

Programs

Haskell

a045980 n = a045980_list !! (n-1)
a045980_list = 0 : filter f [1..] where
   f x = g $ takeWhile ((<= 4 * x) . (^ 3)) $ a027750_row x where
     g [] = False
     g (d:ds) = r == 0 && a010052 (d ^ 2 - 4 * y) == 1 || g ds
       where (y, r) = divMod (d ^ 2 - div x d) 3
-- Reinhard Zumkeller, Dec 20 2013

Mathematica

Union[Select[Sort[Flatten[Table[{j^3-i^3, j^3+i^3}, {i, 0, 20}, {j, i, 20}]]], #<20^3-19^3&]]
With[{nn=20},Take[Union[Select[Flatten[{Total[#],#[[1]]-#[[2]]}&/@(Tuples[ Range[0,nn],2]^3)],#>-1&]],3*nn]] (* Harvey P. Dale, Jun 22 2014 *)

PARI

is(n)=fordiv(n,d, my(L=(d^2-n/d)/3); if(denominator(L)==1 && issquare(d^2-4*L), return(1))); 0 \\ Charles R Greathouse IV, Jun 12 2012

PARI

list(lim)={
    my(v=List(),x3,t);
    for(x=0,sqrtnint(lim\=1,3),
        x3=x^3;
        for(y=0,min(sqrtnint(lim-x3,3),x),
            listput(v,x3+y^3)
        )
    );
    for(x=2,t=sqrtint(lim\3),
        x3=x^3;
        for(y=sqrtnint(max(0,x3-lim-1),3)+1,x-1,
            listput(v,x3-y^3)
        )
    );
    t=(t+1)^3-t^3;
    if(t<=lim,listput(v,t));
    Set(v);
} \\ Charles R Greathouse IV, Jun 12 2012, updated Jan 13 2022

PARI

is(n)=#thue(thueinit(z^3+1),n) \\ Ralf Stephan, Oct 18 2013