1771

1771 = 7 x 11 x 23.

1771 is the smallest of three consecutive numbers that are products of exactly three primes (A113789).

1771 is the 21st tetrahedral number (A000292).

1771 is the concatenation of the 7th prime (17) and its reverse (71) (A067087).

1771 is a divisor of 45⁶ - 1.

1771 is a number n such that n² (3136441) is a concatenation of two nonzero squares (A048375): 3136 = 56² and 441 = 21².

Source: Number Gossip

969

969 = 3 x 17 x 19.

969 and 970 have three distinct prime factors (A140077).

969 is the 17th tetrahedral number (A000292). It is the smallest multidigit tetrahedral palindrome.

969 is a nonagonal number (A001106).

The 18th Mersenne prime has 969 digits (A028335).

The sum of the digits of 969 equals the number of digits times 8 (A061425).

Source: Number Gossip

120

120 = 2 x 2 x 2 x 3 x 5.

120 has a unique representation as a sum of three squares: 120 = 2² + 4² + 10².

120 = 5!

120 is the 15th triangular number. It is the sum of the first 15 positive integers.

120 is the eighth tetrahedral number.

120 is a hexagonal number.

120 = 3¹ + 3² + 3³ + 3⁴.

120 is the smallest composite number of the form x! whose digit sum (1 + 2 + 0 = 3) and x (5) are both primes.

120 is the smallest three-digit number that may be expressed as the difference between squares of consecutive primes in tow different ways: 120 = 31² - 29² = 17² - 13².

The great rhombicosidodecahedron is an Archimedean solid with 120 vertices, 180 edges, and 62 faces.

Source: Prime Curios!

816

816 = 2⁴ x 3 x 17.

816 is the 16th tetrahedral number.

816/(8 x 1 x 6) = 816/48 = 17, a prime. 816 is the largest three-digit integer that has this property.

816 is 30300 in base 4 and 330 in base 16. It is 2244 in base 7 and 424 in base 14.

816 is a telephone area code that covers the Kansas City area in Missouri.

Source: Prime Curios!

286

286 = 2 x 11 x 13.

286 is the 11th tetrahedral number.

286 is a heptagonal number.

286 is the number of rooted trees with 9 vertices.

286 is 2121 in base 5 and 141 in base 15.

286 is the shorthand for the Intel 80286 microprocessor chip.

Source: What's Special About This Number?

680

680 = 2³ x 5 x 17.

680 is the 15th tetrahedral number. It is the smallest tetrahedral number that is also the sum of two tetrahedral numbers: 680 = 560 + 120.

680 has two representations as a sum of two squares: 680 = 2² + 26² = 14² + 22².

680 is 404 in base 13.

680 can be written as the sum of distinct odd powers of two: 680 = 2³ + 2⁵ + 2⁷ + 2⁹.

WCBM AM 680 is a talk radio station in Baltimore.

Source: Number Gossip

455

455 = 5 x 7 x 13.

455 is 111000111 in base 2 (binary), and it is 555 in base 9.

455 is the 13th tetrahedral number.

455 is a number that cannot be written as a sum of three squares.

455 is the number of plus signs needed to write the partitions of 13.

Kathy Mattea sang 455 Rocket, a song celebrating Oldsmobiles of the 1960s and 70s with the high-powered 455 Rocket V8 engine.

Source: Number Gossip

165

165 = 3 x 5 x 11.

165 is 10100101 in base 2 (binary).

165 is the number of integer partitions of 26 into distinct parts.

165 is the ninth tetrahedral number.

165 is the sum of the divisors of the first 14 integers.

The Tahoe Rim Trail is a 165-mile hiking trail around Lake Tahoe.

Source: WolframAlpha

364

364 = 2² x 7 x 13.

364 has a unique representation as a sum of three squares: 364 = 2² + 6² + 18².

364 is the twelfth tetrahedral number.

364 is 111111 in base 3.

The first two digits of 364 are the product of the first two primes to the power of the first prime and the last two digits are the first prime to the product of the first two primes: 36 = (2 x 3)² and 64 = 2^{(2 x 3)}.

Asteroid 364 Isara was discovered in 1893.

Source: Prime Curios!

560

560 = 2⁴ x 5 x 7.

560 has a unique representation as a sum of three squares: 560 = 4² + 12² + 20².

560 is the 14th tetrahedral number. 560 is also an octagonal number.

560 is 202202 in base 3 and 2332 in base 6.

560 is the number of different moves a bishop can make on a chessboard.

Source: Number Gossip

84

84 = 2² x 3 x 7.

84 is tetrahedral number, the sum of the first seven triangular numbers: 84 = 1 + 3 + 6 + 10 + 15 + 21 + 28.

84 has a unique representation as a sum of three squares: 84 = 2² + 4² + 8².

84 is the smallest number with eight representations as a sum of two primes: 84 = 5 + 79 = 11 + 73 = 13 + 71 = 17 + 67 = 23 + 61 = 31 + 53 = 37 + 47 = 41 + 43.

84 is 77 in base 11 and 66 in base 13.

84 Charing Cross Road is a book by Helene Hanff.

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

35

35 = 5 x 7.

35 is the sum of the cubes of the first two primes: 2³ + 3³ = 35.

The product of 35 and 36 has the same prime factors as the product of 4374 and 4375.

35 can be written as a sum of odd primes in 35 ways, and as a sum of primes in 35 x 5 ways.

35 has a unique representation as a sum of three squares: 35 = 1² + 3² + 5².

35 is the smallest number with five representations as a sum of three distinct primes: 35 = 3 + 13 + 19 = 5 + 7 + 23 = 5 + 11 + 19 = 5 + 13 + 17 = 7 + 11 + 17.

35 is the fifth tetrahedral number (the sum of the first five triangular numbers).

35 is the fourth pentatope number.

35 is 55 in base 6.


There are 35 hexominoes, polyominoes made from six squares.


Source: Number Gossip

220

220 = 2² x 5 x 11.

220 is the 10th tetrahedral number.

220 and 284 form the first and smallest pair of amicable numbers. Each is the sum of the aliqot parts of the other. The aliquot parts of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, and 110 and total 284. 284 = 2² x 71 and its aliquot parts are 1, 2, 4, 71, and 142, totalling 220.

220 is a number that cannot be written as a sum of three squares.


In the Bible, 220 was the number of goats Jacob gave to his brother Esau ("two hundred she-goats and twenty he-goats") and the number of sheep ("two hundred ewes and twenty rams," Genesis 32:14).

Source: VirtueScience.com

56

56 = 2³ x 7.

56 is a pronic number because it is the product of two consecutive numbers: 7 x 8 = 56.

56 is the sum of the first six triangular numbers: 1 + 3 + .6 + 10 + 15 + 21 = 56. Hence, it is a tetrahedral number, the number of balls you can put in a triangular pyramid: 1, 4, 10, 20, 35, 56, 84, 120, . . .

56 is the sum of six consecutive prime numbers: 3 + 5 + 7 + 11 + 13 + 17 = 56.


56 men signed the Declaration of Independence in 1776, which led to the formation of the United States of America.

Source: Number Gossip

82

82 = 2² x 3 x 7.

84 has a unique representation as a sum of three squares: 84 = 2² + 4² + 8².

84 is the smallest number with eight representations as a sum of two primes: 84 = 5 + 79 = 11 + 73 = 13 + 71 = 17 + 67 = 23 + 61 = 31 + 53 = 37 + 47 = 41 + 43.

84 is the seventh tetrahedral number.

84 is 77 in base 11.

82 is a Pell-Lucas number (or companion Pell number). The first two numbers in the sequence are both 2, and each successive number is formed by adding twice the previous Pell-Lucas number to the Pell-Lucas number before that. The first few terms of the sequence are 2, 2, 6, 14, 34, 82, 198, 478, . . .

Messier 82 is a starburst galaxy in the constellation Ursa Major.


In both the National Basketball Association and the National Hockey League, each team plays an 82-game regular season schedule.

Source: Wikipedia; On-Line Encyclopedia of Integer Sequences

Image: NASA/JPL-Caltech/STScI/CXC/UofA/ESA/AURA/JHU

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