The paper shows a prime p contributes to the exponent of Cat(n) once for each k with n mod p^k strictly between p^k/2 and p^k-1, a direct consequence of Legendre's formula.
Congruences for Catalan and Motzkin numbers and related sequences
1 Pith paper cite this work. Polarity classification is still indexing.
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abstract
We prove various congruences for Catalan and Motzkin numbers as well as related sequences. The common thread is that all these sequences can be expressed in terms of binomial coefficients. Our techniques are combinatorial and algebraic: group actions, induction, and Lucas' congruence for binomial coefficients come into play. A number of our results settle conjectures of Benoit Cloitre and Reinhard Zumkeller. The Thue-Morse sequence appears in several contexts.
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Factoring Catalan numbers
The paper shows a prime p contributes to the exponent of Cat(n) once for each k with n mod p^k strictly between p^k/2 and p^k-1, a direct consequence of Legendre's formula.