A powerful number is a positive integer m that for every prime number p dividing m, p² also divides m. Powerful numbers are also known as squareful, square-full, or 2-full.

It can be proved that powerful numbers are always of the form a²b³, where a and b both are positive integers. In fact this follows from a simpler arithmetic fact, that the integers

2r+3s

run over all integers >1 as r and s run over integers > 0 (proof by mathematical induction).

The following is a list of all powerful numbers between 1 and 1000:

1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 72, 81, 100, 108, 121, 125, 128, 144, 169, 196, 200, 216, 225, 243, 256, 288, 289, 324, 343, 361, 392, 400, 432, 441, 484, 500, 512, 529, 576, 625, 648, 675, 676, 729, 784, 800, 841, 864, 900, 961, 968, 972, 1000.