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The
Hilbert numbers or
numbers are numbers of the form
.
A016813 .

{1, 5, 9, 13, 17, 21, 25, 29, 33, 37, 41, 45, 49, 53, 57, 61, 65, 69, 73, 77, 81, 85, 89, 93, 97, 101, 105, 109, 113, 117, 121, 125, 129, 133, 137, 141, 145, 149, 153, 157, 161, 165, 169, 173, 177, 181, 185, ...}
Though this set is closed under multiplication (the product of Hilbert numbers is another Hilbert number), it does not form a unique factorization domain^{[1]} (and in any case it does not form a ring as it lacks a zero).^{[2]}
Hilbert primes
The
Hilbert primes or
primes are Hilbert numbers not divisible by any smaller Hilbert number except 1.
A057948 primes: let
S = {1, 5, 9, 13, ..., 4 n + 1, ...}; then an
prime is in
but is not divisible by any members of
except itself and 1.

{5, 9, 13, 17, 21, 29, 33, 37, 41, 49, 53, 57, 61, 69, 73, 77, 89, 93, 97, 101, 109, 113, 121, 129, 133, 137, 141, 149, 157, 161, 173, 177, 181, 193, 197, 201, 209, 213, 217, 229, ...}
Hilbert composites
The
Hilbert composites or
composites are then 25, 45, 65, 81, 85, 105, 117, 125, ... (
A054520 lists the
nonprimes and therefore starts 1, 25, 45, etc.)
A054520 Let
S = {1, 5, 9, 13, ..., 4 n + 1, ...} and call
in
an
prime if
and the only divisors of
in
are 1 and
; sequence gives elements of
that are not
primes.

{1, 25, 45, 65, 81, 85, 105, 117, 125, 145, 153, 165, 169, 185, 189, 205, 221, 225, 245, 261, 265, 273, 285, 289, 297, 305, 325, 333, 345, 357, 365, 369, 377, 385, 405, 425, 429, ...}
Numbers that are Hilbert primes but composites in
are the product of two
primes of the form 4 n + 3 (we can verify that
(4 m + 3) (4 n + 3) = 16 m n + 12 m + 12 n + 9 = 4 s + 1 
; therefore neither prime is a Hilbert number and their product is not divisible by any smaller Hilbert number. These Hilbert primes are listed in
A107978.
Keeping in mind that 9, 21 and 49 are Hilbert primes, we see that 441 does not have a unique factorization into Hilbert primes, since 9 × 49 = 21 2 = 441. See A057949 for more examples.
Notes
 ↑ P. Giblin, Primes and Programming: An Introduction to Number Theory with Computing, Cambridge University Press (1993) p. 30.
 ↑ Weisstein, Eric W., Ring, from MathWorld—A Wolfram Web Resource. (Conditions 3 and 4 are identified as “always required” and both involve 0.)