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How can we find $\sum_{i=1}^n(n\mod i)$?

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This is at the OEIS. Various identities and references are given there.

By calculating a few terms, e.g. the value of your expression when $1 \le n \le 7$ say, you get the sequence 0,0,1,1,4,3,8,8. You can then go to the OEIS and search with those terms, you get the entry linked above.