For a positive integer n, let f(n) be the sum of the squares of the digits (in base 10) of n, e.g.

f(3) = 3² = 9,
f(25) = 2² + 5² = 4 + 25 = 29,
f(442) = 4² + 4² + 2² = 16 + 16 + 4 = 36

Find the last nine digits of the sum of all n, 0 < n < 10²⁰, such that f(n) is a perfect square.