This is more a comment to the question (which I cannot do).

As written (intentionally?) [ADDED: Apparently, it was intentional.] the specified rings are not always the (full) ring of algebraic integers of the field $\mathbb{Q}(\sqrt{c})$ (see, e.g., http://en.wikipedia.org/wiki/Quadratic_integer for details).

In these cases the rings in question are not integrally closed and thus not UFDs, even if the class number of the field is one and thus the (full) ring of algebraic integers would be a UFD (see, e.g., http://en.wikipedia.org/wiki/Class_number_problem ).

Possibly, one needs to take this into account too, when using the list, mentioned in another answer, where the rings are Euclidean.

2 added 159 characters in body

This is more a comment to the question (which I cannot do).

As written (intentionally?) the specified rings are not always the (full) ring of algebraic integers of the field $\mathbb{Q}(\sqrt{c})$ (see, e.g., http://en.wikipedia.org/wiki/Quadratic_integer for details).

In these cases the rings in question are not integrally closed and thus not UFDs, even if the class number of the field is one and thus the (full) ring of algebraic integers would be a UFD (see, e.g., http://en.wikipedia.org/wiki/Class_number_problem ).

Possibly, one needs to take this into account too, when using the list, mentioned in another answer, where the rings are Euclidean.

1

This is more a comment to the question (which I cannot do).

As written (intentionally?) the specified rings are not always the (full) ring of algebraic integers of the field (see, e.g., http://en.wikipedia.org/wiki/Quadratic_integer for details).

In these cases the rings in question are not integrally closed and thus not UFDs, even if the class number of the field is one and thus the (full) ring of algebraic integers would be a UFD (see, e.g., http://en.wikipedia.org/wiki/Class_number_problem ).