There have been numerous papers over the last few years that have been concerned with sequences of integers for which `Kummer-type' Theorems, or `Lucas-type' Theorems or `Wolstenholme-type' Theorems hold. One nice example is the Apéry numbers,

which were introduced in Apéry's proof of the irrationality of

. At first, a few seemingly surprising congruences were found for these numbers, but in 1982, Gessel [7] showed that these were all consequences of the fact that the Apéry numbers satisfy `Lucas--type' and `Wolstenholme--type' Theorems (that is

and

for all

and primes

). R. McIntosh has asked whether a non--trivial sequence of integers, satisfying a `Lucas--type' Theorem, can grow slower than

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