The Catalan numbers are defined by
This note will use results obtained by Liu and Yeh in  to obtain a formula for the number of which are divisible by where
Firstly, as in , let the p-adic order of a positive integer n be defined by
and the cofactor of n with respect to be defined by
In addition the function is defined by
For a number p, we write the base p expansion of a number n as
Then the function is the sum of the p-adic digits of a natural number n (i.e. the sum of the digits of n when n is written in base p). Here we will only be interested in the case . Then
Proof of Theorem 1:
Let . Then
for some arbitrary . Writing in base 2 we have
where there are s 0’s at the end and is arbitrary. So,
where there are s 1’s at the end and is arbitrary. It can be seen that
Since and the possible base 2 representations of n are
with and 1’s at the end.
Therefore, counting each of these possibilities and using the fact that gives
Proof of Corollary 2.
The following result appears in .
Theorem 4 (Corollary 4.3 of ): In general, we have In particular, if and only if , and if and only if
Corollary 2 follows from Theorem 1 and Theorem 4 above since
from Theorem 4
from Theorem 1
Proof of Corollary 3.
From Theorem 4,
from Theorem 1.
 Liu, S.-C., & Yeh, J. (2010). Catalan numbers modulo . J Integer Sequences. Vol 13 (2010) Article 10.5.4