A020894 Nonnegative numbers that are sums of two nonzero cubes.

0, 2, 7, 9, 16, 19, 26, 28, 35, 37, 54, 56, 61, 63, 65, 72, 91, 98, 117, 124, 126, 127, 128, 133, 152, 169, 189, 208, 215, 217, 218, 224, 243, 250, 271, 279, 280, 296, 316, 331, 335, 341, 342, 344, 351, 370, 386, 387, 397, 407, 432, 448, 468, 469

Offset: 1

Author

Steven Finch

Keywords

• nonn

Comments

From Michael B. Porter, Oct 16 2009: (Start)

When calculating terms, there is no need to search beyond a value x defined by x^3 - (x-1)^3 = n. The positive solution is given by x = 1/2 + (sqrt(12n-3))/6.

There are no cubes in this sequence, but the numbers before and after a cube are all included. (End)

Examples

			From _Michael B. Porter_, Oct 16 2009: (Start)
7 is in the sequence because 2^3 + (-1)^3 = 7
8 is not in the sequence because the only solutions to x^3 + y^3 = 8 have either x=0 or y=0. (End)
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
• Steven R. Finch, On a Generalized Fermat-Wiles Equation [broken link]
• Steven R. Finch, On a Generalized Fermat-Wiles Equation [From the Wayback Machine]

Crossrefs

Cf. A045980 [From Michael B. Porter, Oct 16 2009]

Programs

Mathematica

Reap[For[n = 0, n < 500, n++, fi = FindInstance[x > 0 && y != 0 && n == x^3 + y^3, {x, y}, Integers, 1]; If[fi =!= {}, Print[n, " = ", Hold[x^3 + y^3] /. fi[[1]]]; Sow[n]]]][[2, 1]] (* Jean-François Alcover, Nov 05 2016 *)

PARI

isA020894(n) = {r=0;x=1.0/2+sqrt(12*n-3.0)/6;for(i=1,floor(x),if(ispower(n-i^3,3) & (n != i^3),r++));r>0}; \\ Michael B. Porter, Oct 16 2009

PARI

T=thueinit('z^3+1);
is(n)=n==0 || #select(v->v[1] && v[2], thue(T, n))>0 \\ Charles R Greathouse IV, Nov 29 2014

Extensions

Definition and offset edited by N. J. A. Sloane, Dec 01 2009