I'm a totally non maths guy. But I've looked at primes for fun over the last few weeks...
By your definition (not divisible by 2,3,5) I can tell you there are 26 'prime like' numbers between 0-100 28 in 100-200 & 26 between 200-300
What might be useful for you to know is that there are 80 such numbers between 0-300 and then the sequence repeats itself.
In other words in 0-900 there will be 240 'prime like' numbers in 0-90000
there will be (90000/300 x 80) 24000 etc.
In a number not divisible by 300 eg. 1000 just go 1000/300 = 3 remainder 100 so there are 3x80 + 28 (number of 'prime likes' in 0-100)
So 268 'prime like' numbers between 0-1000.
The sequences for your 'very prime like' numbers also start repeating themselves at various intervals and I can give you a cumbersome formula for the answer, but I'm sure an actual maths guy could explain it better.
The above is probably useless to you, just trying to contribute while I wait for someone to answer my post as well.