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

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**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:

“The Oregon [matter] and the annexation of Texas are now all- important to the security and future peace and prosperity of our union, and I hope there are a sufficient *number* of pure American democrats to carry into effect the annexation of Texas and [extension of] our laws over Oregon. No temporizing policy or all is lost.”

—Andrew Jackson (1767–1845)

“The exercise of power is determined by thousands of interactions between the world of the *powerful* and that of the powerless, all the more so because these worlds are never divided by a sharp line: everyone has a small part of himself in both.”

—Václav Havel (b. 1936)