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.. ...

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