Mathematica 7.0.1 says that Sum[1/(k*Length[Divisors[k]]), {k, 1, n}] is the harmonic number $H_n$, which is clearly wrong. The correct answer is at https://mathoverflow.net/questions/18483/an-elementary-number-theoretic-infinite-series

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## Edit:

This is less a bug and more a misunderstanding of how to use *Mathematica*. The culprit is that `Length[Divisors[k]]` for `k` without a value evaluates to `1` (which is consistent with how *Mathematica* structurally treats expressions). The correct way to express the sum is

    Sum[1/(k DivisorSigma[0, k]), {k, 1, n}]

which, as expected, now remains unevaluated.