How to calculate smallest number with certain number of divisors?

Question

From Project Euler problem 500

The number of divisors of 120 is 16. In fact 120 is the smallest number having 16 divisors.

Find the smalle

Answer 1

As Miljen Mikic explains, the divisor counting function is determined by the prime factorisation. To calculate n, start at 1 and use a greedy algorithm to double the number of divisors k times, choosing the cheapest factor at each step. The initial costs are the prime numbers, replaced by their square when you use them. After precomputing the first k prime numbers, you can do this quickly with a min-heap. In Python

import primesieve # pip install primesieve
import heapq

def solve(k, modulus=None):
    """Calculate the smallest number with 2**k divisors."""
    n = 1

    costs = primesieve.generate_n_primes(k) # more than necessary

    for i in range(k):
        cost = heapq.heappop(costs)
        heapq.heappush(costs, cost**2)
        n *= cost
        if modulus:
            n %= modulus

    return n

assert solve(4) == 120

if __name__ == "__main__":
    print(solve(500500, 500500507))