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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?

 Apr 20, 2019
 #1
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+2

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.

 Apr 20, 2019
edited by Guest  Apr 20, 2019
edited by Guest  Apr 20, 2019
edited by Guest  Apr 20, 2019
edited by Guest  Apr 20, 2019
 #2
avatar+103674 
0

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...

 Apr 21, 2019
 #3
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+1

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.

 Apr 21, 2019
 #4
avatar+103674 
+1

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...

----------------

 

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    cool

Melody  Apr 22, 2019
 #5
avatar+178 
+1

Thanks so much Melody and Guests! You really helped me understnad and solve this question!

 Apr 24, 2019

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