Some Hardy Ramanujan variants (Posted on 2007-06-02)

It is well known that 1729 is the smallest integer that can be expressed in two ways as the sum of two perfect cubes: 1729=1³+12³ =9³+10³.

What is the smallest integer that can be expressed in two different ways as the sum of three perfect cubes? And in three different ways as the sum of two perfect cubes?