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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

Assuming that we exclude 1, I get that there are 25 quiteprimes less than 100.  Here's the list:

 

7    11    13    17    19    23    29    31    37    41    43    47    49    53    59    61    67    71    73    77    79    83    89    91    97

 

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