Integers with exactly two non-trivial factors

Having exactly two distinct factors other than 1 and itself means the total number of factors of the integer $N$ is $2 + 2 = 4$.

A number $N$ has 4 factors if:

1. $N = p^3$, where $p$ is a prime.

   • $2^3 = 8 < 50$
   • $3^3 = 27 < 50$
   • $5^3 = 125 > 50$ There are $2$ such numbers ($8, 27$).

2. $N = p_1 \cdot p_2$, where $p_1, p_2$ are distinct primes.

   • With $2$: $2 \times 3 = 6, 2 \times 5 = 10, 2 \times 7 = 14, 2 \times 11 = 22, 2 \times 13 = 26, 2 \times 17 = 34, 2 \times 19 = 38, 2 \times 23 = 46$ (8 numbers)
   • With $3$: $3 \times 5 = 15, 3 \times 7 = 21, 3 \times 11 = 33, 3 \times 13 = 39$ (4 numbers)
   • With $5$: $5 \times 7 = 35$ (1 number) There are $8 + 4 + 1 = 13$ such numbers.

Total positive integers less than 50 with exactly 4 factors $= 2 + 13 = 15$.