1331

1331 is a perfect cube.

1331 = 11³ = (13 – 3 + 1)³ = (1 + 3 x 3 – 1)³.

1331 is that smallest cube (other than 1) containing only odd digits. It is also the smallest cube having every digit it contains exactly twice

1331 = 19² + 21² + 23².

1331 = 11³ = 439 + 443 + 449 is the smallest solution of three consecutive primes summing to the third power of a prime.

A partial solar eclipse occurred on July 5 in the year 1331.

Source: The Penguin Dictionary of Curious and Interesting Numbers, rev. ed., by David Wells.

27

27 = 3³, a perfect cube.

27 is the largest number that is the sum of the digits of its cube: 27³ = 19683; 1 + 9 + 6 + 8 + 3 = 27.

27 is the only number that is three times the sum of its digits.

27 is 11011 in base 2 (binary) and 33 in base 8.

27 is the smallest number with two representations as a sum of three squares: 27 = 1² + 1² + 5² = 3² + 3² + 3².

27 is the number of integer partitions of 15 into distinct parts.

The chemical element cobalt has atomic number 27.

Source: Number Gossip

125

125 is a perfect cube, 5³.

125 is the smallest cube that is also the sum of two squares in two different ways: 125 = 11² + 2² = 10² + 5².

5³ – 2⁷ = 5 – 2. The only known matching pattern is 13³ – 3⁷ = 13 – 3.

125 is the hypotenuse of a primitive Pythagorean triple: 125² = 44² + 117².

Space Shuttle mission STS-125 was the final servicing mission to the Hubble Space Telescope.

Source: The Penguin Dictionary of Curious and Interesting Numbers, rev. ed., by David Wells.