A powerful number (also called a squareful number) is one for which every prime in the factorization appears with exponent at least 2. Equivalently: if a prime \(p\) divides \(n\), then \(p^2\) also divides \(n\).
The first powerful numbers: 1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 72, 81, 100, 108, 121, 125, 128, 144.
Every powerful number can be written uniquely as \(a^2 b^3\) for some positive integers \(a, b\) with \(b\) squarefree. Powerful numbers have density \(\zeta(3)/\zeta(2) \approx 0.7305 \cdot (6/\pi^2)\) — roughly proportional to \(\sqrt{n}\) density up to \(n\), so they thin out quickly.