Counting Divisors
Let $\sigma_0(n)$ be the number of positive divisors of $n$.
For example, $\sigma_0(1) = 1$, $\sigma_0(2) = 2$ and $\sigma_0(6) = 4$.
Let $$S_1(n) = \sum _{i=1}^n \sigma_0(i).$$
Given $N$, find $S_1(N)$.
Input
First line contains $T$ ($1 \le T \le 100000$), the number of test cases.
Each of the next $T$ lines contains a single integer $N$. ($1 \le N < 2^{63}$)
Output
For each number $N$, output a single line containing $S_1(N)$.
Example
Input
5
1
2
3
10
100
Output
1
3
5
27
482
Explanation for Input
- $S_1(3) = \sigma_0(1) + \sigma_0(2) + \sigma_0(3) = 1 + 2 + 2 = 5$
Information
There are 6 input files.
- Input #1: $1 \le N \le 100000$, TL = 2s.
- Input #2: $1 \le T \le 120,\ 1 \le N \le 10^{15}$, TL = 15s.
- Input #3: $1 \le T \le 60,\ 1 \le N \le 10^{16}$, TL = 15s.
- Input #4: $1 \le T \le 25,\ 1 \le N \le 10^{17}$, TL = 15s.
- Input #5: $1 \le T \le 10,\ 1 \le N \le 10^{18}$, TL = 15s.
- Input #6: $1 \le T \le 5,\ 1 \le N < 2^{63}$, TL = 15s.
My C++ solution runs in about 1.3 seconds for each input #2 - #6.
Note
- Probably, $O(\sqrt{n})$ solutions will not pass.
- Intended solutions have a running time of about $O(n^{1/3} \log n)$.
- The answer can be $\ge 2^{64}$.