I'm sure this has been asked before, but how many unique numbers can be made from multiplying $4$ numbers, each between $1$ and $100$?
My guess is the all numbers from $1$ to $100^4$ except those with prime factors above $100$. However this excludes numbers like $11^5$. Then I would also have to exclude numbers with more than $4$ prime factors, and each one is $\ge 11$. I'm probably still missing some though.
Is there a way to find or get an estimate of this number without using a computer? I'm guessing something to do with the prime counting function. Any insight is appreciated.
Edit: Here are some data points (range, unique numbers). Can anyone find a pattern?
10,275 20,2670 30,8679 40,21346 50,49076 60,89247 70,149530 80,253818 90,381413 100,520841