# Definition:Powerful Number

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## Theorem

A **powerful number** is a positive integer such that each of its prime factors appears with multiplicity at least $2$.

That is, each of its prime factors occurs at least squared.

### Sequence

The sequence of powerful numbers begins:

- $1, 4, 8, 9, 16, 25, 27, 32, 36, 49, \ldots$

## Also known as

A **powerful number** can also be referred to as a **squareful**, **square full**, **square-full** or **2-full number**.

## Also see

- Results about
**powerful numbers**can be found here.

## Sources

- 1970: Solomon W. Golomb:
*Powerful Numbers*(*Amer. Math. Monthly***Vol. 77**: pp. 848 – 855) www.jstor.org/stable/2317020

- 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $25$

- Weisstein, Eric W. "Powerful Number." From
*MathWorld*--A Wolfram Web Resource. http://mathworld.wolfram.com/PowerfulNumber.html