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Summing the divisor function d(i) up to n equals counting lattice points under the hyperbola xy โค n, which can be done in O(โn) using floor-division blocks.
Divide and Conquer DP optimization speeds up DP transitions of the form dp[i][j] = min over k of dp[i-1][k] + C(k, j) when the optimal k is monotone in j.