Three random conjectures

Please refute at least one of the following:

\displaystyle \sum_{i=1}^n \dfrac{1}{i} \equiv 0 \pmod{p^2} implies n < p^2

\displaystyle \sum_{i=1}^n \dfrac{1}{i} \equiv 0 \pmod{p^3} implies n = p - 1

\displaystyle \sum_{i=1}^n \dfrac{1}{i} \equiv 0 \pmod{p^4} never happens

EDIT. p = 11, n = 10583 is quite a cute counterexample to the first 2 of these.

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