1175. Prime Arrangements
Description
Return the number of permutations of 1 to n
so that prime numbers are at prime indices (1-indexed.)
(Recall that an integer is prime if and only if it is greater than 1, and cannot be written as a product of two positive integers both smaller than it.)
Since the answer may be large, return the answer modulo 10^9 + 7
.
Example 1:
Input: n = 5 Output: 12 Explanation: For example [1,2,5,4,3] is a valid permutation, but [5,2,3,4,1] is not because the prime number 5 is at index 1.
Example 2:
Input: n = 100 Output: 682289015
Constraints:
1 <= n <= 100
Solutions
Solution 1: Mathematics
First, count the number of prime numbers within the range $[1,n]$, which we denote as $cnt$. Then, calculate the product of the factorial of $cnt$ and $n-cnt$ to get the answer, remember to perform the modulo operation.
Here, we use the "Sieve of Eratosthenes" to count prime numbers.
If $x$ is a prime number, then multiples of $x$ greater than $x$, such as $2x$, $3x$, ... are definitely not prime numbers, so we can start from here.
Let $primes[i]$ indicate whether the number $i$ is a prime number. If it is a prime number, it is $true$, otherwise it is $false$.
We sequentially traverse each number $i$ in the range $[2,n]$. If this number is a prime number, the number of prime numbers increases by $1$, and then all its multiples $j$ are marked as composite numbers (except for the prime number itself), that is, $primes[j]=false$. In this way, at the end of the run, we can know the number of prime numbers.
The time complexity is $O(n \times \log \log n)$.
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