CF1717E Madoka and The Best University

Madoka wants to enter to "Novosibirsk State University", but in the entrance exam she came across a very difficult task: Given an integer $ n $ , it is required to calculate $ \sum{\operatorname{lcm}(c, \gcd(a, b))} $ , for all triples of positive integers $ (a, b, c) $ , where $ a + b + c = n $ . In this problem $ \gcd(x, y) $ denotes the [greatest common divisor](https://en.wikipedia.org/wiki/Greatest_common_divisor) of $ x $ and $ y $ , and $ \operatorname{lcm}(x, y) $ denotes the [least common multiple](https://en.wikipedia.org/wiki/Least_common_multiple) of $ x $ and $ y $ . Solve this problem for Madoka and help her to enter to the best university!

The first and the only line contains a single integer $ n $ ( $ 3 \le n \le 10^5 $ ).

Print exactly one interger — $ \sum{\operatorname{lcm}(c, \gcd(a, b))} $ . Since the answer can be very large, then output it modulo $ 10^9 + 7 $ .

In the first example, there is only one suitable triple $ (1, 1, 1) $ . So the answer is $ \operatorname{lcm}(1, \gcd(1, 1)) = \operatorname{lcm}(1, 1) = 1 $ . In the second example, $ \operatorname{lcm}(1, \gcd(3, 1)) + \operatorname{lcm}(1, \gcd(2, 2)) + \operatorname{lcm}(1, \gcd(1, 3)) + \operatorname{lcm}(2, \gcd(2, 1)) + \operatorname{lcm}(2, \gcd(1, 2)) + \operatorname{lcm}(3, \gcd(1, 1)) = \operatorname{lcm}(1, 1) + \operatorname{lcm}(1, 2) + \operatorname{lcm}(1, 1) + \operatorname{lcm}(2, 1) + \operatorname{lcm}(2, 1) + \operatorname{lcm}(3, 1) = 1 + 2 + 1 + 2 + 2 + 3 = 11 $