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:
“Love is at the root of all healthy discipline. The desire to be loved is a powerful motivation for children to behave in ways that give their parents pleasure rather than displeasure. it may even be our own long-ago fear of losing our parents’ love that now sometimes makes us uneasy about setting and maintaining limits. We’re afraid we’ll lose the love of our children when we don’t let them have their way.”
—Fred Rogers (20th century)
“No Government can be long secure without a formidable Opposition. It reduces their supporters to that tractable number which can be managed by the joint influences of fruition and hope. It offers vengeance to the discontented, and distinction to the ambitious; and employs the energies of aspiring spirits, who otherwise may prove traitors in a division or assassins in a debate.”
—Benjamin Disraeli (1804–1881)