Sum of Least Common Multiples

Given a positive integer \(n\), calculate the following sum:

[ S = \sum_{i = 1}^{n} \operatorname{lcm}(i, n), ]

where \(\operatorname{lcm}(i, n)\) denotes the least common multiple of \(i\) and \(n\). The answer should be computed exactly as defined above.

inputFormat

The input consists of a single integer \(n\) (\(1 \leq n \leq 10^5\)).

outputFormat

Output a single integer representing \(S\), the sum of \(\operatorname{lcm}(i, n)\) for all \(1 \leq i \leq n\).

sample

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