  A formula on Stirling numbers of the second kind $S(n, k)$ is proved. As a
corollary, for odd $n$ and even $k$, it is shown that $k!S(n, k)$ is a positive
multiple of the greatest common divisor of $j!S(n, j)$ for $k+1\leq j\leq n$.
Also, as an application to algebraic topology, some isomorphisms of unstable
$K^1$-groups of stunted complex projective spaces are deduced.
