Many numbers can be expressed as the sum of a square and a cube. Some of them in more than one way.
Consider the palindromic numbers that can be expressed as the sum of a square and a cube, both greater than $1$, in exactly $4$ different ways.
For example, $5229225$ is a palindromic number and it can be expressed in exactly $4$ different ways:
$2285^2 + 20^3$
$2223^2 + 66^3$
$1810^2 + 125^3$
$1197^2 + 156^3$
Find the sum of the five smallest such palindromic numbers.
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