Powerful Number

A powerful number is a positive integer m such that for every prime number p dividing m, p2 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 = a2b3, 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 NumberEquivalence 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:

    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)