OFFSET

1,1

COMMENTS

I have verified that for all integers n <= 10000, there always exist triples (x,y,z) of positive integers such that x^3 + n*x*y + y^3 = z^3.

For example: n=4459, (2835, 14700, 15015) is a solution.

EXAMPLE

a(3)=26 because there exists a triple (14,24,26) satisfying 14^3 + 3*14*24 + 24^3 = 26^3, and no smaller solution exists.

MATHEMATICA

Table[First[

Sort@Flatten[

Table[CubeRoot[x^3 + n*x*y + y^3], {x, 1, 400}, {y, x + 1, 400}]],

IntegerQ[#] &], {n, 1, 30}]

PROG

(Python)

def a(n):

r=400

for x in range(1, 400):

for y in range(x+1, 400):

zz=x**3+n*x*y+y**3

z=round(zz**(1/3))

if zz==z**3 and z<r:

r=z

return(r)

print([a(n) for n in range(1, 31)])

CROSSREFS

KEYWORD

nonn

AUTHOR

Zhining Yang, Mar 11 2023

EXTENSIONS

More terms from Sean A. Irvine, Apr 10 2023

STATUS

approved