  Amdeberhan conjectured that the number of $(t,t+1, t+2)$-core partitions is
$\sum_{0\leq k\leq [\frac{t}{2}]}\frac{1}{k+1}\binom{t}{2k}\binom{2k}{k}$. In
this paper, we obtain the generating function of the numbers $f_t$ of $(t, t +
1, ..., t + p)$-core partitions. In particular, this verifies that Amdeberhan's
conjecture is true. We also prove that the number of $(t_1,t_2,..., t_m)$-core
partitions is finite if and only if gcd$(t_1,t_2,..., t_m)=1,$ which extends
Anderson's result on the finiteness of the number of $(t_1,t_2)$-core
partitions for coprime positive integers $t_1$ and $t_2$ and thus rediscover a
result of Keith and Nath with a different proof.
