## APS2 - Amazing Prime Sequence (hard)

This problem is a harder version of APS.

Let $f(n)$ be the smallest prime factor of $n$. For example, $f(2) = 2,\ f(4) = 2$ and $f(35) = 5$.

The sequence $S(n)$ is defined for all positive integers as follows:

• $S(1) = 0$
• $S(n) = S(n-1) + f(n)$ (if $n \ge 2$)

Given $N$, find $S(N)$ modulo $2^{64}$.

### Input

First line contains $T$ ($1 \le T \le 10000$), the number of test cases.

Each of the next $T$ lines contains a single integer $N$. ($1 \le N \le 1234567891011$)

### Output

For each integer $N$, output a single line containing $S(N)$ modulo $2^{64}$.

### Example

#### Input

51410010000001000000000000

#### Output

071257375684049897294954823202325427

### Explanation for Input

- $S(4) = 2 + 3 + 2 = 7$

- $S(10^{12}) = 18435592284459044389811 \equiv 7294954823202325427 \pmod{2^{64}}$

### Information

There are 6 Input files.

- Input #0: $1 \le T \le 10000$, $1 \le N \le 10000$, TL = 1s.

- Input #1: $1 \le T \le 1000$, $1 \le N \le 10^{8}$, TL = 20s.

- Input #2: $1 \le T \le 200$, $1 \le N \le 10^{9}$, TL = 20s.

- Input #3: $1 \le T \le 40$, $1 \le N \le 10^{10}$, TL = 20s.

- Input #4: $1 \le T \le 7$, $1 \le N \le 10^{11}$, TL = 20s.

- Input #5: $T = 1$, $1 \le N \le 1234567891011$, TL = 20s.

My solution runs in 5.36 sec. (total time)

Source Limit is 8 KB.