Problem of the Week

The famous harmonic sequence {a₁ ,a₂ ,a₃ a_4,...} has the terms

It's well-known that the terms of this sequence grow without bound, so the terms will eventually exceed any given positive integer.  (But don't hold your breath as the terms grow excruciatingly slowly -- for instance, a_k first exceeds 10 at k = 12367, and at k = 1,000,000 you haven't yet reached 15.)

Clearly a₁ is an integer. Find all values of k for which a_k is an integer.

(For the lionhearted:  Find all integers that can be expressed as a_n - a_m, the difference of two terms of the harmonic sequence, or equivalently, find all integers which can be expressed as the sum of the reciprocals of consecutive integers.)

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ã2002 Alberto L. Delgado