It is a prime number, its only divisor being 1. It is also the 7th number in the Fibonacci sequence.
13 is equal to the sum of the squares of the digits of its own square in bases 4 and 83.
13 is a happy number: https://en.wikipedia.org/wiki/Happy_number
Triskaidekaphobia - fear of the number 13. First found reference to the triskaidekaphobia dates to the late 1800s. The term triskaidekaphobia is derived from Ancient Greek τρεισκαίδεκα (treiskaídeka, “thirteen”) and φόβος (phóbos, "aversion"/"fear")
Many multiple story buildings do not contain the 13th floor as superstition.
A number is divisible by the number 13 if that if you add 4 times the last digit to the rest, the result must be divisible by 13.
13 is the smallest emirp. 13 is prime, and its reversal, 31, is also prime.