The Motzkin numbers are defined by
where are the Catalan numbers. The following result is established in 
Theorem 1 (Theorem 5.4 of ). The Motzkin number is divisible by if and only if is one of the following forms
where and .
Theorem 2: The asymptotic density of numbers of the first form in theorem 1 is . Numbers of the fourth form also have asymptotic density . The asymptotic density of numbers of the second and third forms in theorem 1 is each.
Corollary 3: The asymptotic density of is .
Remark 4: Corollary 3 follows immediately from theorems 1 and 2 and the disjointness of the 4 forms of integers listed in theorem 1.
Remark 5: Numerical tests also show that roughly 22.5% of Motzkin numbers are congruent to each of .
Proof of Theorem 2.
Firstly consider numbers of the form . As we are interested in asymptotic density it is enough to look at numbers of the form . We have, for fixed ,
where is an error term introduced by not rounding down to the nearest integer. So,
where the new error term satisfies . Then
A similar argument can be used to establish the asymptotic densities of the other 3 forms of numbers in the theorem.
 Deutsch, E., & Sagan, B. E. (2006). Congruences for Catalan and Motzkin numbers and related sequences. Journal of Number Theory, 117(1), 191–215. http://doi.org/10.1016/j.jnt.2005.06.005