In practice, the set of powerful numbers consists of the number 1 plus all numbers in whose factorizations the primes appears with exponents greater than 1. This set coincides with the set of numbers of the form , for .
There are infinite pairs of consecutive powerful numbers, the smallest being (8, 9), but Erdös, Mollin, and Walsh conjectured that there are no three consecutive powerful numbers.
Heath-Brown has shown in 1988 that every sufficiently large natural number is the sum of at most three powerful numbers. Probably the largest number which is not the sum of 3 powerful numbers is 119.
The sum of the reciprocals of the powerful numbers converges to .
P.T.Bateman & E.Grosswald have proved that the asymptotic number of powerful numbers up to is given by