Tough-to-test composites

Let a tough-to-test composite be a positive integer that is composite, but not divisible by 2, nor 3, nor 5.

Given that there are 168 prime numbers between 1 and 1000, how many tough-to-test composite numbers are there between 1 and 1000?

Notes: 1 is neither prime nor composite. There are some tough-to-test composite numbers that many would recognize as composite. For example, 49 and 77.

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