Double Summation with Coprime Indicator

Given a positive integer \(n\), compute the value of the double summation:

\( S = \sum_{i=1}^n \sum_{j=1}^n \left\lfloor \frac{n}{\max(i,j)} \right\rfloor \; [i \perp j] \)

where \([i \perp j]\) is an indicator that equals 1 if \(i\) and \(j\) are coprime (i.e. \(\gcd(i,j) = 1\)) and 0 otherwise.

Hint: This problem suggests the use of advanced number theory techniques such as Möbius inversion. In the process of optimizing complex summations related to divisors, the Möbius inversion formula is a key tool.

inputFormat

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

outputFormat

Output a single integer, the computed value of the summation \(S\).

sample

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