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

1,3

Comments

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

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

Kevin A. Broughan, Characterizing the sum of two cubes, J. Integer Seqs., Vol. 6, 2003.

M. Kim, Diophantine equations in two variables

Index entries for sequences related to sums of cubes

Example

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

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 *)

Prog

(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
(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

Crossrefs

A004999 and A003325 are subsequences.

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

Sequence in context: A037455 A020675 A317303 * A104339 A199004 A168064

Adjacent sequences: A045977 A045978 A045979 * A045981 A045982 A045983

Keyword

nonn,easy,nice

Author

N. J. A. Sloane

Status

approved