A085705 Integers expressible as (x^3 + y^3 + z^3)/(x*y*z) with nonzero integers x, y and z. Alternatively, integers expressible as a/b + b/c + c/a with nonzero integers a, b and c.

3, 5, 6, 9, 10, 13, 14, 15, 16, 17, 18, 19, 20, 21, 26, 29, 30, 31, 35, 36, 38, 40, 41, 44, 47, 51, 53, 54, 57, 62, 63, 64, 66, 67, 69, 70, 71, 72, 73, 74, 76, 77, 83, 84, 86, 87, 92, 94, 96, 98, 99, 101, 102, 103, 105, 106, 107, 108, 109, 110, 112, 113, 116

Offset

1,1

Comments

A representation k = (x^3 + y^3 + z^3)/(x*y*z) is equivalent to 2 representations k = a/b + b/c + c/a, given by a=y^2*z, b=z^2*x, c=x^2*y and a=y*z^2, b=x*y^2, c=z*x^2. - Dean Hickerson, Jul 14 2003

For each a(n) > 5 there are infinitely many representations. - David J. Rusin, Jul 15 2003

Links

Jinyuan Wang, Table of n, a(n) for n = 1..1000

David J. Rusin, For which values of n is a/b + b/c + c/a = n solvable? [Broken link]

David J. Rusin, For which values of n is a/b + b/c + c/a = n solvable? [Cached copy of html wrapper for paper but in pdf format (so none of the links work)]

David J. Rusin, For which values of n is a/b + b/c + c/a = n solvable? [Cached copy of .txt file]

Example

15 is in the sequence because 15 = (7^3 + (-3)^3 + (-1)^3)/(7*-3*-1) = (7^3 - 3^3 - 1^3)/(7*3*1) = (343 - 27 - 1)/21. This is equivalent to 15 = -9/7 - 7/147 + 147/9 or 15 = -3/63 - 63/49 + 49/3.
16 = (70^3 + (-31)^3 + (-9)^3)/(70*-31*-9) = (70^3 - 31^3 - 9^3)/(70*31*9) = (343000 - 29791 - 729)/19530.

Prog

(PARI) is(k) = abs(k-4)==1 || ellanalyticrank(ellinit([0, k^2, 0, -72*k, -16*(4*k^3+27)]))[1]; \\ Jinyuan Wang, Jul 27 2021

Crossrefs

Cf. A072716 (representation by positive x, y, z).

Sequence in context: A018900 A126590 A140584 * A187417 A072716 A167384

Adjacent sequences: A085702 A085703 A085704 * A085706 A085707 A085708

Keyword

nonn

Author

Hugo Pfoertner, Jul 18 2003

Extensions

More terms from David J. Rusin, Jul 26 2003

Status

approved