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:
“Throughout the 1980’s, we did hear too much about individual gain and the ethos of selfishness and greed. We did not hear enough about how to be a good member of a community, to define the common good and to repair the social contract. And we also found that while prosperity does not trickle down from the most powerful to the rest of us, all too often indifference and even intolerance do.”
—Hillary Rodham Clinton (b. 1947)
“I who have been involved with all styles of painting can assure you that the only things that fluctuate are the waves of fashion which carry the snobs and speculators; the number of true connoisseurs remains more or less the same.”
—Pablo Picasso (1881–1973)