180 (number)

In mathematics

180 is an abundant number, with its proper divisors summing up to 366. 180 is also the 11th highly composite number, a positive integer with 18 divisors, more than any smaller positive integer. One of the consequences of 180 having so many divisors is that it is a practical number, meaning that any positive number smaller than 180 that is not a divisor of 180 can be expressed as the sum of some of 180's divisors. 180 is a Harshad number and a refactorable number.

180 is the sum of two square numbers: 122 + 62. It can be expressed as either the sum of six consecutive prime numbers: 19 + 23 + 29 + 31 + 37 + 41, or the sum of eight consecutive prime numbers: 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37. 180 is an Ulam number, which can be expressed as a sum of earlier terms in the Ulam sequence only as 177 + 3.

180 is a 61-gonal number, while 61 is the 18th prime number.

Half a circle has 180 degrees, and thus a U-turn is also referred to as a 180.

Summing Euler's totient function φ(x) over the first + 24 integers gives 180.

In binary it is a digitally balanced number, since its binary representation has the same number of zeros as ones (10110100).

A triangle has three interior angles that collectively total 180 degrees. In general, the interior angles of an n-sided polygon add to (n-2) \times 180 degrees.

References

References

1. "Positive Integers: 180".
2. "The Number 180". VirtueScience.com.
3. "A002182 - OEIS".
4. {{Cite OEIS. A002182. Highly composite numbers
5. (2016-04-18). "Refactorable numbers". The OEIS Foundation.
6. (2016-04-18). "Ulam numbers". The OEIS Foundation.
7. Wells, D.. (1987). "[[The Penguin Dictionary of Curious and Interesting Numbers]]". Penguin Group.