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 Number: Equivalence of The Two Definitions, Mathematical Properties, Sums and Differences of Powerful Numbers, Generalization
Famous quotes containing the words powerful and/or number:
“For each illness that doctors cure with medicine, they provoke ten in healthy people by inoculating them with the virus that is a thousand times more powerful than any microbe: the idea that one is ill.”
—Marcel Proust (1871–1922)
“Mining today is an affair of mathematics, of finance, of the latest in engineering skill. Cautious men behind polished desks in San Francisco figure out in advance the amount of metal to a cubic yard, the number of yards washed a day, the cost of each operation. They have no need of grubstakes.”
—Merle Colby, U.S. public relief program (1935-1943)