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?

Rudram592 Apr 20, 2019

#1**+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.**

Guest 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

edited by Guest Apr 20, 2019

edited by Guest Apr 20, 2019

edited by Guest Apr 20, 2019

#2**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...

Melody Apr 21, 2019

#3**+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.

Guest Apr 21, 2019

#4**+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...

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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 **

Melody
Apr 22, 2019