题解 | #求幂#

import java.util.HashSet;
import java.util.Scanner;

/**
 * JD14 求幂
 * @author d3y1
 */
public class Main {
    private static final int MOD = 1000000007;

    public static void main(String[] args){
        Scanner in = new Scanner(System.in);

        while(in.hasNext()){
            solution(in);
        }
    }

    /**
     * 模拟法: 数学
     *
     * 举例子找规律
     * 两种情况:
     * 1. a=c时
     * (1) a=c=1
     * 式子个数: n*n
     * (2) a=c>1 (2<=a=c<=n)
     * 式子个数: n*(n-1)
     *
     * 合计: n*n+n*(n-1) = 2*n^2-n
     *
     * 2. a!=c时
     * a^b = c^d
     * => (i^x)^b = (i^y)^d
     * 遍历i, 且i^x<=n i^y<=n, 求出x和y的范围
     * 遍历x和y, 求出x和y的最大公约数m(m=gcd(x,y)), 放到底数做i的指数, (i^m)^n1*b = (i^m)^n2*d
     * 转换成求 n1*b = n2*d 有多少种情况
     * x和y的较大值除以最大公约数m得到n1(n1=max(x,y)/m)
     * b=(n2/n1)*d (n1>n2 b<=n d<=n)
     * 可见构造d为n1的倍数即可 共有n/n1个
     * 等式两边可以对调 结果*2
     *
     * 合计: (n/n1)*2 = (n/(max(x,y)/m))*2 = (n/(max(x,y)/gcd(x,y)))*2
     *
     * @param in
     */
    private static void solution(Scanner in){
        int n = in.nextInt();
        // n^2+n*(n-1) = 2*n^2-n
        long result = (2L*n*n-n)%MOD;

        HashSet<Integer> set = new HashSet<>();
        for(int i=2; i*i<=n; i++){
            if(set.contains(i)){
                continue;
            }

            int base = i;
            int pow = 0;
            while(base <= n){
                set.add(base);
                pow++;
                base *= i;
            }

            for(int x=1; x<=pow; x++){
                for(int y=x+1; y<=pow; y++){
                    result = (result+(n/(y/gcd(x,y)))*2)%MOD;
                }
            }
        }

        System.out.println(result);
    }

    /**
     * 最大公约数
     * @param a
     * @param b
     * @return
     */
    private static int gcd(int a, int b){
        return b>0?gcd(b,a%b):a;
    }
}