1. For any given natural number $n$, if $n^{6}+3 a$ is a positive integer, then
(A) there are infinitely many such $a$;
(B) such $a$ exist, but there are only finitely many;
(C) such $a$ do not exist;
(D) none of the conclusions (A), (B), (C) are correct.
1. (A)

For any natural number $k$, take
$$
\begin{array}{c}
a=3 n^{4} k+9 n^{2} k^{2}+9 k^{2}, \\
n^{6}+3 a=\left(n^{2}+3 k\right)^{3} .
\end{array}
$$