the original question asked for what is: ? perhaps the difficuly is in what you meant by "it" in your post...your "it" and Pinkk's "it" may be 2 different things.
your orignal post (#11) was clearer.
i think this is an interesting problem, which "as-stated" doesn't have a definitive answer. i haven't looked very far into it, but my feeling is, that if p is a divisor of the "denominator" for a closed form expression of , each power will resolve to 1, and we get p-1. the "denominators" in question are the even-numbered terms in this sequence:
A064538 - OEIS
which are also the denominators of the even Bournoulli numbers, expressed in "lowest terms". it seems to me that this question is a "hard" question, which is belied by its simple statement.
the complete answer of this question, then, rests on proving the following:
let k be n even integer, and let denote the k-th Bernoulli number. if p divides the denominator of written in lowest terms, then
conjecture: the statement "p divides the denominator..." can be replaced by p-1 divides k.