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:

“In a *number* of other cultures, fathers are not relegated to babysitter status, nor is their ability to be primary nurturers so readily dismissed.... We have evidence that in our own society men can rear and nurture their children competently and that men’s methods, although different from those of women, are imaginative and constructive.”

—Kyle D. Pruett (20th century)

“Women of my age in America are at the mercy of two *powerful* and antagonistic traditions. The first is the ultradomestic fifties with its *powerful* cult of motherhood; the other is the strident feminism of the seventies with its attempt to clone the male competitive model.... Only in America are these ideologies pushed to extremes.”

—Sylvia Ann Hewitt (20th century)