Some arithmetical properties of Apéry numbers (Q1917897)

The Apéry numbers are defined by \[ a_n= \sum^n_{k=0} {n\choose k}^2 \cdot {n+k \choose k}^2. \] In analogy with the well known Lucas congruence for binomial coefficients, the author proves that for any positive integer \(n\) and any prime \(p\geq 5\) we have \[ a_n \equiv \prod^{t(r)}_{i=0} a_{n_i(r)} \pmod {p^r} \quad \text{if} \quad n= \sum^{t(r)}_{i =0} n_i(r)(p^r)^i \] is the expansion of \(n\) in base \(p^r\) \((r\in \{1, 2, 3\})\). The proof is based mainly on the binomial congruence \({np\choose ip} \equiv {n\choose i} \pmod {p^3}\), with \(n\), \(i\) positive integers, \(p\geq 5\) primes. Extensive numerical tables on the factors of \(a_n\) are also listed.