  Simultaneous core partitions have attracted much attention since Anderson's
work on the number of $(t_1,t_2)$-core partitions. In this paper we focus on
simultaneous core partitions with distinct parts. The generating function of
$t$-core partitions with distinct parts is obtained. We also prove the results
on the number, the largest size and the average size of $(t, t + 1)$-core
partitions. This gives a complete answer to a conjecture of Amdeberhan, which
is partly and independently proved by Straub, Nath and Sellers, and Zaleski
recently.
