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1808. Maximize Number of Nice Divisors

Description

You are given a positive integer primeFactors. You are asked to construct a positive integer n that satisfies the following conditions:

• The number of prime factors of n (not necessarily distinct) is at most primeFactors.
• The number of nice divisors of n is maximized. Note that a divisor of n is nice if it is divisible by every prime factor of n. For example, if n = 12, then its prime factors are [2,2,3], then 6 and 12 are nice divisors, while 3 and 4 are not.

Return the number of nice divisors of n. Since that number can be too large, return it modulo 109 + 7.

Note that a prime number is a natural number greater than 1 that is not a product of two smaller natural numbers. The prime factors of a number n is a list of prime numbers such that their product equals n.

 

Example 1:

Input: primeFactors = 5
Output: 6
Explanation: 200 is a valid value of n.
It has 5 prime factors: [2,2,2,5,5], and it has 6 nice divisors: [10,20,40,50,100,200].
There is not other value of n that has at most 5 prime factors and more nice divisors.

Example 2:

Input: primeFactors = 8
Output: 18

 

Constraints:

• 1 <= primeFactors <= 109

Solutions

Solution 1: Problem Transformation + Fast Power

We can factorize $n$ into prime factors, i.e., $n=a_1^{k_1}\times a_2^{k_2}\times \cdots \times a_m^{k_m}$, where $a_i$ is a prime factor and $k_i$ is the exponent of the prime factor $a_i$. Since the number of prime factors of $n$ does not exceed primeFactors, we have $k_1+k_2+\cdots +k_m\le primeFactors$.

According to the problem description, we know that a good factor of $n$ must be divisible by all prime factors, which means that a good factor of $n$ needs to include $a_1\times a_2\times \cdots \times a_m$ as a factor. Then the number of good factors $k=k_1\times k_2\times \cdots \times k_m$, i.e., $k$ is the product of $k_1,k_2,\cdots ,k_m$. To maximize the number of good factors, we need to split primeFactors into $k_1,k_2,\cdots ,k_m$ to make $k_1\times k_2\times \cdots \times k_m$ the largest. Therefore, the problem is transformed into: split the integer primeFactors into the product of several integers to maximize the product.

Next, we just need to discuss different cases.

• If $primeFactors<4$, then directly return primeFactors.
• If $primeFactors$ is a multiple of $3$, then we split primeFactors into multiples of $3$, i.e., $3^{\frac{primeFactors}{3}}$.
• If $primeFactors$ modulo $3$ equals $1$, then we split primeFactors into $\frac{primeFactors}{3}-1$ multiples of $3$, and then multiply by $4$, i.e., $3^{\frac{primeFactors}{3}-1}\times 4$.
• If $primeFactors$ modulo $3$ equals $2$, then we split primeFactors into $\frac{primeFactors}{3}$ multiples of $3$, and then multiply by $2$, i.e., $3^{\frac{primeFactors}{3}}\times 2$.

In the above process, we use fast power to calculate the modulus.

The time complexity is $O(\log n)$, and the space complexity is $O(1)$.

• Java
• C++
• Python
• Go
• Javascript

• class Solution {
      private final int mod = (int) 1e9 + 7;
  
      public int maxNiceDivisors(int primeFactors) {
          if (primeFactors < 4) {
              return primeFactors;
          }
          if (primeFactors % 3 == 0) {
              return qpow(3, primeFactors / 3);
          }
          if (primeFactors % 3 == 1) {
              return (int) (4L * qpow(3, primeFactors / 3 - 1) % mod);
          }
          return 2 * qpow(3, primeFactors / 3) % mod;
      }
  
      private int qpow(long a, long n) {
          long ans = 1;
          for (; n > 0; n >>= 1) {
              if ((n & 1) == 1) {
                  ans = ans * a % mod;
              }
              a = a * a % mod;
          }
          return (int) ans;
      }
  }
  

• class Solution {
  public:
      int maxNiceDivisors(int primeFactors) {
          if (primeFactors < 4) {
              return primeFactors;
          }
          const int mod = 1e9 + 7;
          auto qpow = [&](long long a, long long n) {
              long long ans = 1;
              for (; n; n >>= 1) {
                  if (n & 1) {
                      ans = ans * a % mod;
                  }
                  a = a * a % mod;
              }
              return (int) ans;
          };
          if (primeFactors % 3 == 0) {
              return qpow(3, primeFactors / 3);
          }
          if (primeFactors % 3 == 1) {
              return qpow(3, primeFactors / 3 - 1) * 4L % mod;
          }
          return qpow(3, primeFactors / 3) * 2 % mod;
      }
  };
  

• class Solution:
      def maxNiceDivisors(self, primeFactors: int) -> int:
          mod = 10**9 + 7
          if primeFactors < 4:
              return primeFactors
          if primeFactors % 3 == 0:
              return pow(3, primeFactors // 3, mod) % mod
          if primeFactors % 3 == 1:
              return 4 * pow(3, primeFactors // 3 - 1, mod) % mod
          return 2 * pow(3, primeFactors // 3, mod) % mod
  
  

• func maxNiceDivisors(primeFactors int) int {
  	if primeFactors < 4 {
  		return primeFactors
  	}
  	const mod = 1e9 + 7
  	qpow := func(a, n int) int {
  		ans := 1
  		for ; n > 0; n >>= 1 {
  			if n&1 == 1 {
  				ans = ans * a % mod
  			}
  			a = a * a % mod
  		}
  		return ans
  	}
  	if primeFactors%3 == 0 {
  		return qpow(3, primeFactors/3)
  	}
  	if primeFactors%3 == 1 {
  		return qpow(3, primeFactors/3-1) * 4 % mod
  	}
  	return qpow(3, primeFactors/3) * 2 % mod
  }
  

• /**
   * @param {number} primeFactors
   * @return {number}
   */
  var maxNiceDivisors = function (primeFactors) {
      if (primeFactors < 4) {
          return primeFactors;
      }
      const mod = 1e9 + 7;
      const qpow = (a, n) => {
          let ans = 1;
          for (; n; n >>= 1) {
              if (n & 1) {
                  ans = Number((BigInt(ans) * BigInt(a)) % BigInt(mod));
              }
              a = Number((BigInt(a) * BigInt(a)) % BigInt(mod));
          }
          return ans;
      };
      const k = Math.floor(primeFactors / 3);
      if (primeFactors % 3 === 0) {
          return qpow(3, k);
      }
      if (primeFactors % 3 === 1) {
          return (4 * qpow(3, k - 1)) % mod;
      }
      return (2 * qpow(3, k)) % mod;
  };
  
  

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