A Theorem involving the denominators of Bernoulli numbers
Consider the average of the first n k-th powers. We pose and answer the following natural question: For which values of n and k is this average an integer? If k is odd the answer is easy; it is an integer as long as n is incongruent to 2 modulo 4. If k is even then the criterion involves the denominator of the k-th Bernoulli number. The average is an integer iff $n$ is not divisible by any prime which divides the denominator of the k-th Bernoulli number.
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