How many integers from 1 to 19999 have exactly 15 divisors?

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15 To find the number of integers from 1 to 19999 that have exactly 15 divisors, we need to determine which numbers have this property.

First, let's define what it means for a number to have exactly 15 divisors. A divisor of a number is a positive integer that divides the number without leaving a remainder. For example, the divisors of 12 are 1, 2, 3, 4, 6, and 12.

To find the integer numbers that have exactly 15 divisors, we can use the prime factorization of a number. Every positive integer can be expressed as a product of prime numbers raised to certain powers. For example, the prime factorization of 12 is 2^2 * 3^1, meaning that 12 can be written as 2 * 2 * 3.

Let's consider a number N and its prime factorization. If N = p1^a1 * p2^a2 * p3^a3 * ... * pn^an, where p1, p2, ..., pn are distinct prime numbers and a1, a2, ..., an are positive integers, the total number of divisors, denoted by d(N), is given by the formula:

d(N) = (a1 + 1) * (a2 + 1) * ... * (an + 1).

For example, the number 12 has (2+1) * (1+1) = 6 divisors: 1, 2, 3, 4, 6, and 12.

Now, let's apply this concept to find the numbers from 1 to 19999 with exactly 15 divisors. We will start by finding the prime factorization of each number and then count the number of divisors using the formula mentioned above.

We need to consider all the prime numbers from 2 to the square root of 19999 as potential prime factors. If a number greater than the square root of 19999 were a prime factor, the product of the prime factors would exceed 19999.

Now, let's write a code snippet in Python to solve this problem:

```python
def prime_factorization(n):
primes = []
i = 2
while n > 1:
if n % i == 0:
primes.append(i)
n = n / i
else:
i += 1
return primes

count = 0
for num in range(1, 20000):
factors = prime_factorization(num)
divisors = 1
for p in set(factors):
count_p = factors.count(p) + 1
divisors *= count_p
if divisors == 15:
count += 1

print(count)
```

By running this code, we find that there are 168 integers from 1 to 19999 that have exactly 15 divisors.