- 38! − 1 yields 523022617466601111760007224100074291199999999, which is the 16th factorial prime.
- There is no answer to the equation φ(x) = 38, making 38 a nontotient.
- 38 is the sum of the squares of the first three primes.
- 37 and 38 are the first pair of consecutive positive integers not divisible by any of their digits.
- 38 is the largest even number which cannot be written as the sum of two odd composite numbers.
- There are only two normal magic hexagons, order 1 (which is trivial) and order 3. The sum of each row of an order 3 magic hexagon is 38.