I will guide you on how to solve it for the first two, that is your "quiteprimes" under 100 and under 1000.
1) Numbers under 100:
Since you are not allowed to divide by primes LESS THAN 2, 3, and 5, that means you must consider prime factors of 7 and higher. As a consequence, you must divide 100 / 7 =floor(14). Then you must count ALL prime number between 7 and 14. And you have:7, 11 and 13 =3. Next:floor(100/11)=9 and 9 < 11, therefore there are no prime to consider.
So, the total number of "quiteprimes" is the number of prime numbers as we calculated them above =3
2)Numbers under 1000
This is very similar to above calculation for 100, except for the larger number of 1000:
You will divide: 1000 / 7=floor(142). Then you will have to count all prime number between 7 and 142 and you should get: 31. Next you will divide 1000 / 11 =floor(90). Then you will count all prime numbers between 11 and 90 and you should get:20. Then you would continue with this process for: 1000 / 13 =floor(76) and count all the prime numbers beteen 13 and 76 and you should get:16.......and so on. Then you would add them all up:31 + 20 + 16.......etc. and you should get: 94.
3) All other examples in your question are calculated in EXACTLY the same way, except for a big number like 90,000 it will take quite a bit of work to count them all.
Note: There maybe an easier way of calculating them, but I don't know of any. Good luck to you.