Powerful numbers, definition (1): if a prime p divides n then p^2 must also divide n (also called squareful, square full, square-full or 2-powerful numbers)

A001694 on the OEIS · family multiplicative · also known as Powerful numbers

Weight–level plate of Powerful numbers, definition (1): if a prime p divides n then p^2 must also divide n (also called squareful, square full, square-full or 2-powerful numbers) — Click the plate to explore it in 3-D. Weight k across, level L up, both on log scales: blue in the weight class (k ≤ L), orange in the level class (k > L). The dashed diagonal is k = L.

Open in the 3-D viewer A001694 on the OEIS

Terms	100,000 (n = 1 … 100,000)
Decomposable (a > 2d)	99,996
Level class, k > L	71,900 · 71.90 %
Weight class, k ≤ L	28,096 · 28.10 %
Ties, k = L	1
On the level line L = 1	4,053
Forced level, l ≤ d²	42,065
Range of a(n)	1 … 2,200,079,025
Range of the jump d	1 … 93,809
Largest weight k, level L	2,199,703,799, 147,044,169

p | n implies p^2 | n; each term is a^2 b^3 with b squarefree, uniquely. count(x) ~ (zeta(3/2)/zeta(3)) sqrt(x) = 2.17 sqrt(x), so the sequence thins out like a quadratic one. The jump outgrows sqrt(l) on 42.07 % of terms (forced level) and 71.90 % are level-classified.

The decomposition

Every term of a strictly increasing sequence is written a(n) = k(n)·L(n) + d(n): the jump d = a(n+1) − a(n), the weight k the least divisor of a − d greater than d, the level L = (a − d)/k. A term decomposes when a > 2d; it is in the level class when k > L and in the weight class when k ≤ L. See decompwlj.com and arXiv:0711.0865, or how it works, with worked examples and a live one.