421

421 is a prime number.

419 and 421 are twin primes.

421 is a centered square number.

421 is the smallest number formed by the powers of two in logical order from right to left.

421 is 111 in base 20.

421 is the sum of five consecutive prime numbers: 421 = 73 + 79 + 83 + 89 + 97.

421 has a representation as a sum of two squares: 421 = 14² + 15².

421 is the hypotenuse of a primitive Pythagorean triple: 421² = 29² + 420².

All integers ever tested in the Collatz problem eventually end in the sequence 4-2-1.

Source: Prime Curios!

761

761 is a prime number. It is a Sophie Germain prime because 2 x 761 + 1 = 1523 is also prime.

761 is a centered square number.

761 is the sum of all primes numbers up to 79.

761 has a representation as a sum of two squares: 761 = 19² + 20².

761 is the hypotenuse of a primitive Pythagorean triple: 761² = 39² + 760².

A sequence of six nines (known as the Feynman point) begins immediately after the 761st decimal place of pi. Physicist Richard Feynman expressed a wish to memorize the digits of pi as far as that point so that when reciting them he would be able to end with ". . . nine, nine, nine, nine, nine, nine, and so on."

Source: Prime Curios!

365

365 = 5 x 73.

365 is 101101101 in base 2 (binary). It is 555 in base 8.

365 is a centered square number and a centered tridecagonal number (A051865).

365 is the smallest number that can be written as a sum of consecutive squares in more than one way: 365 = 14² + 13² = 12² + 11² + 10².

365 has two representations as a sum of two squares: 365 = 2² + 19² = 13² + 14².

365 is the hypotenuse of two primitive Pythagorean triples: 365² = 27² + 364² = 76² + 357².

365 is the most frequent number of days in a year.

Source: What's Special About This Number?

841

841 = 29 x 29.

841 is a centered square number. It is also a centered heptagonal number and a centered octagonal number.

841 has a representation as a sum of two squares: 841 = 20² + 21². It is the sum of two consecutive squares.

841 is the hypotenuse of a primitive Pythagorean triple: 841² = 41² + 840².

841 is the smallest 3-digit square whose digits are decreasing.

841 is the smallest binary prime square. It's a perfect square whose binary representation (1101001001) considered as a decimal is a prime number.

841 is a divisor of 41⁴ - 1.

Source: Number Gossip

113

113 is a prime number. It is a Sophie Germain prime because 2 x 113 + 1 = 227 is also a prime.

113 is the smallest three-digit prime whose product and sum of digits is prime: 1 x 1 x 3 = 3 and 1 + 1 + 3 = 5.

113 has a representation as a sum of two squares: 113 = 7² + 8² (the sum of two consecutive squares).

113 is the hypotenuse of a primitive Pythagorean triple: 113² = 15² + 112².

113 is a centered square number.

113² = 12769 and its reversal 96721 = 311².

A113 is the license plate number of Andy's mom's vehicle in the film Toy Story.

Source: Number Gossip and Prime Curios!

613

613 is a prime number.

613 is a centered square number.

613 is the smallest prime such that bringing the first digit to the back produces a triangular number (136) and repeating this operation produces a square (361).

613 is the smallest number n such that the numbers of divisors of n, n + 1, n + 2, and n + 3 are k, 2k, 4k, and 8k for k = 2.

613 has a representation as a sum of two squares: 613 = 17² + 18².

613 is the hypotenuse of a primitive Pythagorean triple: 613² = 35² + 612².

The 613 commandments (613 Mitzvot) is a numbering of the statements and principles of law, ethics, and spiritual practice contained in the Torah (the Five Books of Moses).

Source: Number Gossip

962

962 = 2 x 13 x 37.

962 has two representations as a sum of two squares: 962 = 1² + 31² = 11² + 29².

962 is 4242 in base 6 and 282 in base 20.

962 is the sum of three non-zero fourth powers: 962 = 3⁴ + 4⁴ + 5⁴.

962 is a centered square number: 4n² + 4n + 2, for n = 15.

The Porsche 962 was a sports-prototype racing car, first built in 1984.

Source: On-Line Encyclopedia of Integer Sequences

545

545 = 5 x 109.

545 is a centered square number.

545 is the smallest palindrome resulting from the sum of an ascending sequence of consecutive primes with a descending sequence of the same primes as exponents: 2⁷ + 3⁵ + 5³ + 7² = 545.

1*1! + 2*2! + 3*3! + . . . + 545*545! is a titanic prime.

545 has a base 3 representation (202012) that begins with its base 4 representation (20201).

545 people are said to govern the United States: 435 representatives, 100 senators, 1 president, and 9 justices of the Supreme Court.

Source: Prime Curios!