Finding solutions for a given pair (number of factors, number of prime factors)

Question

I have been given x and k, where x is the number of factors of a number A, and k is the number of prime fact

Answer 1

First your implementation is too inefficient

• Don't call a function inside the for loop like this

  for (long long int numbers = 1; numbers < pow(10, 9) && !stop; numbers++)
  

  pow will be called unnecessarily on each iteration. Save every constant to a variable before the loop. But in this case just use numbers < 1e9

• To get all factors of n you just need to loop until sqrt(n). You're doing for (long long int a = 1; a <= numbers && !stop; a++) { if (numbers % a == 0) { so for example instead of only 10⁴ loops for 10⁸ you'll need 10⁸ loops. If numbers % a == 0 then both a and numbers/a are factors of numbers

But the real issue here is that your algorithm is flawed. You can get all factors of A if you know the prime factors of A and their exponents. If A = p₁^m p₂ⁿ ... p_k^p then A will have the following factors:

• p₁ⁱ for i = 1..m
• p₂ⁱ for i = 1..n
• ...
• p_kⁱ for i = 1..p
• Factors from 2 prime factors: p₁ⁱp₂^j, p₁ⁱp₃^j,... p₁ⁱp_k^j, p₂ⁱp₃^j, p₂ⁱp₄^j,... p₂ⁱp_k^j,... p_k-1ⁱp_k^j
• Factors from 3 prime factors...
• Factors from k prime factors

This is purely a combination counting issue that can be done easily without even knowing A. Notice that the number of factors from k prime factors has some relation to the number of factors from k-1 prime factors