  Johnson recently proved Armstrong's conjecture which states that the average
size of an $(a,b)$-core partition is $(a+b+1)(a-1)(b-1)/24$. He used various
coordinate changes and one-to-one correspondences that are useful for counting
problems about simultaneous core partitions. We give an expression for the
number of $(b_1,b_2,\cdots, b_n)$-core partitions where
$\{b_1,b_2,\cdots,b_n\}$ contains at least one pair of relatively prime
numbers. We also evaluate the largest size of a self-conjugate
$(s,s+1,s+2)$-core partition.
