Expectation Division

Time Limit: 6000/3000 MS (Java/Others)    Memory Limit: 131072/131072 K (Java/Others)
Total Submission(s): 0    Accepted Submission(s): 0
Special Judge

Problem Description

To be frank with you, this problem is a classic problem of tremendous magnitude which may increase the difficulty of this problem.

We define a type of operation concerning a positive integer n

(n>1)

as to replace it with an integer d

, one of factors of n

(1≤d≤n)

.

You are given a positive integer n

and then we will ask you to determine the expectation number of times to utilize this type of operation if we want to change n

into 1

by operating again and again, assuming each possible d

in each operation has equal possibility to select.

For the sake of calculation, n

and all its distinct prime factors p1,p2,?,pm

will be given, satisfying n

has m

distinct prime factors exactly.

Input

The input contains multiple test cases.

For each test case:

The first line contains two positive integers n

and m

which indicates m

is the number of distinct prime factors of n

, satisfying 2≤n≤1024

.

The second lines contains m

distinct prime numbers p1,p2,?,pm

, satisfying 2≤pi≤106

.

About 2⋅105

test cases in total.

Warm Tips for C/C++: __int128_t is available here but standard solutions of this problem do not use this compiler-dependent data type.

Output

For each test case, output "Case #x

" in one line (without quotes), where x

indicates the case number starting from 1

and y

denotes the expectation number of times to utilize this type of operation of corresponding case. Your answer will be considered correct if its absolute or relative error won‘t exceed 10−9

.

Sample Input

2 1
2
4 1
2
6 2
2 3
8 1
2
10 2
2 5
12 2
2 3

Sample Output

Case #1: 2.0000000000
Case #2: 2.5000000000
Case #3: 2.6666666667
Case #4: 2.8333333333
Case #5: 2.6666666667
Case #6: 3.0333333333