60

This number is a composite.

The largest integer such that all smaller integers relatively prime to it are either a prime, a power of a prime, or 1. [Noe]

60 is the smallest number which is the sum of two odd primes in six different ways. [Wolman]

The smallest composite number which is the order of a simple group, namely A5 (the alternating group on 5 letters). [Terr]

The largest known positive integer that cannot be written as p * q + r, where p, q, r are three distinct primes. [Capelle]

60 is the only known number that is the common pair-sum of two different first runs of consecutive primes with symmetrical gaps about the center: the first run of 8 (17...43) and the first run of 10 (13...47). Both runs are symmetric about the primorial 30. [Ahmed]