GOTW- Counting - Page 16

I'm not even gonna post a numberrr!! hahahha 😆 fine 119

120 is the factorial of 5, and the sum of a twin prime pair (59 + 61). 120 is the sum of four consecutive prime numbers (23 + 29 + 31 + 37), four consecutive powers of 2 (8+16+32+64), and four consecutive powers of 3 (3 + 9 + 27 + 81). It is highly composite, superabundant, and colossally abundant number, with its 16 divisors being more than any number lower than it has, and it is also the smallest number to have exactly that many divisors. It is also a sparsely totient number. 120 is the smallest number to appear six times in Pascal's triangle. 120 is also the smallest multiple of 6 with no adjacent prime number.

It is the eighth hexagonal number and the fifteenthtriangular number, as well as the sum of the first eight triangular numbers, making it also a tetrahedral number. 120 is divisible by the first 5 triangular numbers and the first 4 tetrahedral numbers.

120 is the first multiply perfect number of order three (a 3-perfect number, triperfect). The sum of its factors (including one and itself) sum to 360; exactly three times 120. Note that perfect numbers are order two (2-perfect) by the same definition.

120 is divisible by the number of primes below it, 30 in this case. However there is no integer which has 120 as the sum of its proper divisors, making 120 an untouchable number.

The sum of Euler's totient function f(x) over the first nineteen integers is 120.

120 figures in Pierre de Fermat's modified Diophantine problem as the largest known integer of the sequence 1, 3, 8, 120. Fermat wanted to find another positive integer that multiplied with any of the other numbers in the sequence yields a number that is one less than a square. Leonhard Euler also searched for this number, but failed to find it, but did find a fractional number that meets the other conditions, 777480 / 2879².

The internal angles of a regular hexagon (one where all sides and all angles are equal) are all 120 degrees.

120 is a Harshad number in base 10.

One hundred [and] twenty-one is a square and is the sum of three consecutive primes (37 + 41 + 43). There are no squares besides 121 known to be of the form , where p is prime (3, in this case). Other such squares must have at least 35 digits.

There are only two other squares known to be of the form n! + 1, supporting Brocard's conjecture. Another example of 121 being of the few examples supporting a conjecture is that Fermat conjectured that 4 and 121 are the only perfect squares of the form x³ - 4 (with x being 2 and 5, respectively).^[1]

It is also a star number and a centered octagonal number.

A Chinese checkers board has 121 holes

In base 10, it is a Smith number since its digits add up to the same value as its factorization (which uses the same digits) and as a consequence of that it is a Friedman number (11^2). But it can not be expressed as the sum of any other number plus that number's digits, making 121 a self number.

Originally posted by: -Nakshatra-

All cheaters club 😡