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Try thinking in this direction

S = sum(a[i]*i*i); where i=1..sqrt(10 000 000), a[i] are multipliers. Objective function min(sum(a[i]))

It's clear that the longest variant is the sum of ones. Taking ones as a base, you can reduce the objective function by adding squares of larger numbers. And when iterating, don't consider variants where the objective function takes a larger value than the current one.

The idea is that you need to collapse ones to minimize the objective function. This is one possible direction to move in. If anyone has something better, write to me, I'm also interested.

Here is [something] that proves the following fact.

If x^2 + y^2 = n, then

(x+y)^2 + (x-y)^
                            
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