  In this paper, we study $(s,s+1)$-core partitions with $d$-distinct parts. We
obtain results on the number and the largest size of such partitions, so we
extend Xiong's paper in which the results are obtained about $(s,s+1)$-core
partitions with distinct parts. Also, we propose a conjecture about
$(s,s+r)$-core partitions with $d$-distinct parts for $1 \le r \le d$.
