A **powerful number** is a positive integer *m* such that for every prime number *p* dividing *m*, *p*2 also divides *m*. Equivalently, a powerful number is the product of a square and a cube, that is, a number *m* of the form *m* = *a*2*b*3, where *a* and *b* are positive integers. Powerful numbers are also known as **squareful**, **square-full**, or **2-full**. Paul Erdős and George Szekeres studied such numbers and Solomon W. Golomb named such numbers *powerful*.

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 (sequence A001694 in OEIS).

Read more about Powerful Number: Equivalence of The Two Definitions, Mathematical Properties, Sums and Differences of Powerful Numbers, Generalization

### Other articles related to "powerful number, number, powerful numbers, powerful":

**Powerful Number**- Generalization

... Such an integer is called a k-

**powerful number**, k-ful

**number**, or k-full

**number**... (2k+1 − 1)k, 2k(2k+1 − 1)k, (2k+1 − 1)k+1 are k-

**powerful numbers**in an arithmetic progression ... as are k-

**powerful**in an arithmetic progression with common difference d, then a1(as + d)k, a2(as + d)k.. ...

### Famous quotes containing the words number and/or powerful:

“How often should a woman be pregnant? Continually, or hardly ever? Or must there be a certain *number* of pregnancy anniversaries established by fashion? What do you, at the age of forty-three, have to say on the subject? Is it a fact that the laws of nature, or of the country, or of propriety, have ordained this time of life for sterility?”

—Philip Dormer Stanhope, 4th Earl Chesterfield (1694–1773)

“Nearly all our *powerful* men in this age of the world are unbelievers; the best of them in doubt and misery; the worst of them in reckless defiance; the plurality in plodding hesitation, doing, as well as they can, what practical work lies ready to their hands.”

—John Ruskin (1819–1900)