A181123 Numbers that are the differences of two positive cubes.

0, 7, 19, 26, 37, 56, 61, 63, 91, 98, 117, 124, 127, 152, 169, 189, 208, 215, 217, 218, 271, 279, 296, 316, 331, 335, 342, 386, 387, 397, 448, 469, 485, 488, 504, 511, 513, 547, 602, 604, 631, 657, 665, 702, 721, 728, 784, 817, 819, 866, 875, 919, 936, 973

Offset

1,2

Comments

Because x^3-y^3 = (x-y)(x^2+xy+y^2), the difference of two cubes is a prime number only if x=y+1, in which case all the primes are cuban, see A002407.

The difference can be a square (see A038597), but Fermat's Last Theorem prevents the difference from ever being a cube. Beal's Conjecture implies that there are no higher odd powers in this sequence.

If n is in the sequence, it must be x^3-y^3 where 0 < y <= x < n^(1/2). - Robert Israel, Dec 24 2017

Links

T. D. Noe and Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 1000 from Noe)

Maple

N:= 10^4: # to get all terms <= N
sort(convert(select(`<=`, {0, seq(seq(x^3-y^3, y=1..x-1), x=1..floor(sqrt(N)))}, N), list)); # Robert Israel, Dec 24 2017

Mathematica

nn=10^5; p=3; Union[Reap[Do[n=i^p-j^p; If[n<=nn, Sow[n]], {i, Ceiling[(nn/p)^(1/(p-1))]}, {j, i}]][[2, 1]]]
With[{nn=60}, Take[Union[Abs[Flatten[Differences/@Tuples[ Range[ nn]^3, 2]]]], nn]] (* Harvey P. Dale, May 11 2014 *)

Prog

(PARI) list(lim)=my(v=List([0]), a3); for(a=2, sqrtint(lim\3), a3=a^3; for(b=if(a3>lim, sqrtnint(a3-lim-1, 3)+1, 1), a-1, listput(v, a3-b^3))); Set(v) \\ Charles R Greathouse IV, Jan 25 2018

Crossrefs

Cf. A014439, A014440, A014441, A333376, A333377, A038593, A086120.

Cf. A024352 (squares), A147857 (4th powers), A181124-A181128 (5th to 9th powers).

Sequence in context: A055246 A003282 A006063 * A038593 A014439 A342160

Adjacent sequences: A181120 A181121 A181122 * A181124 A181125 A181126

Keyword

nonn

Author

T. D. Noe, Oct 06 2010

Status

approved