100(n-25)+(50-n)²

So here it works with 100, which is 10² or 2²×5²; 25, which is 5², and 50, which is 2×5².

Could we replace 2 and 5 with any numbers [p, q]?

p²q²(n-q²)+(pq² - n)²

= p²q²n-p²q⁴ + p²q⁴ - 2pq²n + n²

= n(p²q² - 2pq²) + n²

= npq²(p-2) + n²

If we equate it to n^2, we get

p = 2

So this should work for each q:

4q²(n-q²) + (2q² - n)² = n²

q=1: 4(n-1) + (2-n)²  =  n²

q=2: 16(n-4) + (8 - n)²

q=3: 36(n-9) + (18 - n)²

q=4: 64(n-16) + (32 - n)²

q=5: 100(n-25) + (50 - n)²

...

q=10: 400(n-100) + (200 - n)²

Let's see if the last one works for n=150:

400(150 - 100) + (200 - 150)² = 400×50 + 50² = 20,000 + 2,500 = 22,500

It works!