19600

19600 = 140², a perfect square.

19600 = 2⁴ x 5² x 7².

19600 is one of only three numbers that are simultaneously square and tetrahedral. Tetrahedral numbers correspond to the number of balls in a triangular pyramid in which each side contains n + 1 balls. They are the sum of consecutive triangular numbers. 19600 is the 48th tetrahedral number.

19600 =

140²

= 1 + 3 + 6 + 10 + 15 + . . . + 1176.

The others are 1 = 1² = 1 and 4 = 2² = 1 + 3.

A triangular pyramid with side length 5 contains 35 spheres. Each layer represents one of the first five triangular numbers.

19600 has a representation as a sum of two squares: 19600 = 84² + 112².

A contestant in Nathan’s Famous Hot Dog Eating Contest in New York who consumes 66 hot dogs (the world record) would ingest a total of 19,600 calories.

Sources: Wolfram MathWorld; Wikipedia; CalorieLab