Answer: 15, 16, 17, 18, 19
Let the five consecutive natural numbers be \(x - 2, x - 1, x, x + 1, x + 2\).
\((x-2)^2 + (x-1)^2 + x^2 + (x+1)^2 + (x+2)^2 = 1455\)
\(5x^2 + 10 = 1455\)
\(5x^2 = 1445 \Rightarrow x^2 = 289 \Rightarrow x = 17\) (x is a natural number).
The numbers are 15, 16, 17, 18, 19.
Check: \(225 + 256 + 289 + 324 + 361 = 1455\) ✓.