# Numbers not Expressible as Sum of Less than 9 Positive Cubes

The following are the only positive integers cannot be expressed as the sum of less than $9$ positive cubes:
 $\displaystyle 23$ $=$ $\displaystyle 2^3 + 2^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3$ $\displaystyle 239$ $=$ $\displaystyle 4^3 + 4^3 + 3^3 + 3^3 + 3^3 + 3^3 + 1^3 + 1^3 + 1^3$