729

729 = 3 x 3 x 3 x 3 x 3 x 3 = 3⁶. It is a perfect sixth power.

729 is a centered octagonal number.

728 and 729 are consecutive Smith numbers, so they are known as Smith brothers.

729 is the smallest odd cube represented as the sum of two prime numbers: 729 = 2 + 727.

729 is a cube that is the sum of three cubes: 729 = 9³ = 1³ + 6³ + 8³.

729 = 27² = (1 + 2 + 2 + 2 + 4 + 4 + 4 + 8)² = 1³ + 2³ + 2³ + 2³ + 4³ + 4³ + 4³ + 8³.

729 is 1331 in base 8 and 121 in base 26.

Source: Prime Curios!

625

625 = 5⁴. It is a perfect fourth power.

Two fourth powers use the same digits: 5⁴ = 625 and 4⁴ = 256.

625 has two representations as a sum of two squares: 625 = 7² + 24² = 15² + 20².

625 is the hypotenuse of a primitive Pythagorean triple: 625² = 336² + 527².

625 is a centered octagonal number.

625 is an automorphic number.

According to a 2010 White House news release, 625 people lost their health insurance every hour in 2009.

Source: What's Special About This Number?

64

64 = 2⁶.

64 is the first number after 1 to be both a square and a cube: 64 = 8² = 4³.

64 is the number of integer partitions of 20 into distinct parts.

64 is 121 in base 7 and 44 in base 15. It is 1000000 in base 2, 1000 in base 4, 100 in base 8, 40 in base 16, and 20 in base 32.

The Commodore 64 was an 8-bit home computer introduced in 1982.

Source: Phillips, R. 1994. Number Facts, Figures and Fiction. Cambridge University Press.

81

81 = 3⁴, a perfect fourth power.

81 is the only integer other than 1 that is the square of the sum of its digits.

81 is a heptagonal number.

81 is a centered octagonal number.

81 = 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 (the first nine odd numbers).

The chemical element thallium has the atomic number 81.

Source: Number Gossip

243

243 = 3⁵, a perfect fifth power.

243 is the first number of the form p^q where p and q are a twin prime pair (3, 5).

243 is the sum of the first three palindromic primes: 243 = 11 + 101 + 131.

243 is 3303 in base 4, 1433 in base 5, 1043 in base 6, 363 in base 8, 300 in base 9, 183 in base 12, and 113 in base 15.

243 Ida is a large, irregularly shaped asteroid. It was the second asteroid to be visited by a spacecraft and the first to be found to possess a satellite.

Source: Number Gossip

16 777 216

16777216 = 2²⁴, a perfect 24th power.

16777216 = 8⁸. 1¹, 8⁸, and 9⁹ are the only powers that have the same number of decimal digits as the power.

16777216 (8⁸) is the number of labeled mappings from n points to themselves, for n = 8.

16777216 is the smallest number beginning with 1 and having exactly n prime divisors (counted with multiplicity), for n = 24.

Computer bug report: Fatal error: Allowed memory size of 16777216 bytes exhausted.

In the RGB color model, the Truecolor scheme defines 256 shades of red, blue, and green, which results in 16,777,216 color variations for each pixel (256 x 256 x 256).

Source: Wikipedia

512

512 = 2⁹.

512 has a representation as a sum of two squares: 512 = 16² + 16².

512 is the number of integer partitions of 34 into distinct parts.

512 is the smallest integer (except 1) that is the cube of the sum of its digits: 512 = (5 + 1 + 2)³.

512 is 1000000000 in base 2, 20000 in base 4, 1331 in base 7, 1000 in base 8, 242 in base 15, and 200 in base 16.

512 is the three-digit start in the Dewey Decimal Classification System of call numbers for books in algebra and number theory.

Source: Number Gossip