A080761 Positive numbers of the form y^2 - x^3, x and y >= 1.

1, 3, 8, 9, 12, 15, 17, 18, 19, 22, 24, 28, 30, 35, 36, 37, 38, 40, 41, 44, 48, 54, 55, 56, 57, 63, 64, 65, 68, 71, 73, 79, 80, 89, 92, 94, 97, 98, 99, 100, 101, 105, 106, 107, 108, 112, 113, 117, 119, 120, 121, 128, 129, 131, 132, 136, 138, 141, 142, 143, 145, 148, 151

Offset

1,2

Comments

From Artur Jasinski, Oct 03 2007: (Start)

Some numbers have multiple partitions:

8 = 4^2 - 8^3 = 312^2 - 46^3,

9 = 6^2 - 3^3 = 15^2 - 6 ^3 = 253^2 - 40^3. (End)

This is Mordell's equation with the condition that x and y are positive. Sequence A054504 lists the n for which there is no solution to Mordell's equation. Hence, none of those numbers will be in this sequence. The terms of this sequence can be determined by looking at the link to Gebel's data. - T. D. Noe, Mar 23 2011

Links

Table of n, a(n) for n=1..63.

J. Gebel, Integer points on Mordell curves [Cached copy, after the original web site tnt.math.se.tmu.ac.jp was shut down in 2017]

J. Gebel, A. Petho and H. G. Zimmer, On Mordell's equation, Compositio Math. 110 (1998), 335-367.

Cino Hilliard, Proof that n^3+7 <> k^2 for all integers n,k.

Example

8 is in the sequence since 3^2 = 1^3 + 8.

Mathematica

With[{nn=100}, Take[Union[Select[First[#]^2-Last[#]^3&/@Tuples[Range[ 20nn], 2], #>0&]], nn]] (* Harvey P. Dale, Jul 10 2012 *)

Prog

(PARI) diop(n, m) = { for(p=1, m, for(x=1, n, y=x*x*x+p; if(issquare(y), print1(p" "); break) ) ) }

Crossrefs

Complement of A080762.

Cf. sequences for n^3+7, n^3+17, n^3+3, n^3+2, n^3+5.

Cf. A029727, A029728, A134042, A134043.

Sequence in context: A028960 A139491 A084387 * A087286 A165289 A066494

Adjacent sequences: A080758 A080759 A080760 * A080762 A080763 A080764

Keyword

nonn

Author

Cino Hilliard, Mar 10 2003

Extensions

"Positive" added to definition by N. J. A. Sloane, Oct 06 2007

Status

approved