+218 Call a positive integer kinda-prime if it has a prime number of positive integer divisors. If there are 168 prime numbers less than 1000, how many kinda-prime positive integers are there less than 1000?
I count 17 such numbers:
2^2, 3^2, 3^3, 4^2, 5^2, 7^2, 8^2, 9^2, 11^2, 13^2, 17^2, 19^2, 23^2, 25^2, 27^2, 29^2, 31^2.
Note: Of course, I did not include any of the 168 primes, since all of them have 2 divisors, which of course is a prime number.
+118687 Call a positive integer kinda-prime if it has a prime number of positive integer divisors. If there are 168 prime numbers less than 1000, how many kinda-prime positive integers are there less than 1000?
Mmm
Prime numbers between 1 and 500 (from quora.com)
1–100 -> 25 numbers (2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97)
101–200 -> 21 numbers (101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 183, 193, 197, 199)
201–300 -> 16 numbers (211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293)
301–400 -> 16 numbers (307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397)
401–500 -> 17 numbers (401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499)
Beats me...
Melody: He/she is NOT concerned about prime numbers themselves. Rather, he/she is asking "numbers whose positive integer DIVISORS TOTAL is a prime number." Example: 4^2 =16 =1, 2, 4, 8, 16 (5 divisors). It is "5 divisors" which is a Prime Number. At least, that is how I read the question.
+118687 Thanks guest,
I did understand that.
Those numbers are how many factors any kinda-prime number less than 1000 can have.
I have no idea where to go from there.
You have collected some of them but I suspect there are many more. Umm...
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All factors are in pairs so all prime numbers have 2 factors, 2 is a prime number, so all prime numbers are sorta-prime as well.
There are 168 of those (could have worked it our using Sieve of Eratosthenes but I let someone else on the net do that for me)
Certainly at least some of the squared numbers work.
How about 10^2 ? 1 2 4 5 10 20 25 50 100 That has 9 factors so that is no good.
Squares of primes would all work becasue they have 3 factors: that is these ones squared 2,3,5,7,9,11,13,17,19,23,29,31, (that is 12)
Cubes of primes have 4 factors so they are no good.
primes to the power of 4 have 5 factors so they are all good. 2,3,4,5 (that is 3)
Primes to the power of 6 have 7 factors so they are all good 2 and 3 (that is 2)
So now I have 168 + 12+3+2 = 168+ 17
that is the same as what you got guest.
I can't think of any others either.
You did well guest 
