CF1542C.Strange Function

普及/提高-

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Let $f(i)$ denote the minimum positive integer $x$ such that $x$ is not a divisor of $i$ .

Compute $\sum_{i=1}^n f(i)$ modulo $10^9+7$ . In other words, compute $f(1)+f(2)+\dots+f(n)$ modulo $10^9+7$ .

The first line contains a single integer $t$ ( $1\leq t\leq 10^4$ ), the number of test cases. Then $t$ cases follow.

The only line of each test case contains a single integer $n$ ( $1\leq n\leq 10^{16}$ ).

For each test case, output a single integer $ans$ , where $ans=\sum_{i=1}^n f(i)$ modulo $10^9+7$ .

输入#1

6
1
2
3
4
10
10000000000000000

输出#1

In the fourth test case $n=4$ , so $ans=f(1)+f(2)+f(3)+f(4)$ .

$1$ is a divisor of $1$ but $2$ isn't, so $2$ is the minimum positive integer that isn't a divisor of $1$ . Thus, $f(1)=2$ .
$1$ and $2$ are divisors of $2$ but $3$ isn't, so $3$ is the minimum positive integer that isn't a divisor of $2$ . Thus, $f(2)=3$ .
$1$ is a divisor of $3$ but $2$ isn't, so $2$ is the minimum positive integer that isn't a divisor of $3$ . Thus, $f(3)=2$ .
$1$ and $2$ are divisors of $4$ but $3$ isn't, so $3$ is the minimum positive integer that isn't a divisor of $4$ . Thus, $f(4)=3$ .

Therefore, $ans=f(1)+f(2)+f(3)+f(4)=2+3+2+3=10$ .