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:
“...A shadow now occasionally crossed my simple, sanguine, and life enjoying mind, a notion that I was never really going to accomplish those powerful literary works which would blow a noble trumpet to social generosity and noblesse oblige before the world. What? should I find myself always planning and never achieving ... a richly complicated and yet firmly unified novel?”
—Sarah N. Cleghorn (1876–1959)
“In many ways, life becomes simpler [for young adults]. . . . We are expected to solve only a finite number of problems within a limited range of possible solutions. . . . It’s a mental vacation compared with figuring out who we are, what we believe, what we’re going to do with our talents, how we’re going to solve the social problems of the globe . . .and what the perfect way to raise our children will be.”
—Roger Gould (20th century)