The sum of a Hilbert number amount of Hilbert numbers (1 number, 5 numbers, 9 numbers, etc.) is also a Hilbert number.
Hilbert primes
A Hilbert prime is a Hilbert number that is not divisible by a smaller Hilbert number (other than 1). The sequence of Hilbert primes begins
5, 9, 13, 17, 21, 29, 33, 37, 41, 49, ... (sequence A057948 in the OEIS).
A Hilbert prime is not necessarily a prime number; for example, 21 is a composite number since 21 = 3 ⋅ 7. However, 21 is a Hilbert prime since neither 3 nor 7 (the only factors of 21 other than 1 and itself) are Hilbert numbers. It follows from multiplication modulo 4 that a Hilbert prime is either a prime number of the form 4n + 1 (called a Pythagorean prime), or a semiprime of the form (4a + 3) ⋅ (4b + 3).
References
Flannery, S.; Flannery, D. (2000), In Code: A Mathematical Journey, Profile Books