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Do numbers only divisible by 1, themselves, and their square roots have a specific term?

Seems like they're all squares of primes: 1, 9, 25, 49, 121, 169... I feel like I'm missing something really obvious, I apologize if that's the case. Searching using the obvious keywords didn't bring anything directly referencing this class of numbers to light.

primeroot
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  • $1$ is not a square of a prime. Obviously these numbers must be square themselves, can you prove that a square of a composite number does not have the property you seek? – player3236 Mar 26 '21 at 02:51
  • OEIS is always a good place to look https://oeis.org/A001248 ... $p_n^2$ ? – Donald Splutterwit Mar 26 '21 at 02:54
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    there are collective terms such as "prime powers" and "prime squares." – Will Jagy Mar 26 '21 at 03:03
  • @player3236 I don't care about proving anything, just looking for a name for the class of numbers that strictly complies with the rules I stated – primeroot Mar 26 '21 at 16:41
  • @WillJagy If an example of a prime power is 64 (2^6), that class of numbers does not conform to the rules I mentioned – primeroot Mar 26 '21 at 16:45
  • I meant "prime squares" for you. I use the phrase; there may or may not be anyone else who uses it. For me, it is a useful phrase relating to binary quadratic forms with integer coefficients – Will Jagy Mar 26 '21 at 16:48
  • @WillJagy I agree it's useful, it was the first thing I thought of. But since I didn't find anything which specifically defined it like this I wasn't sure if there was another term in use that I was missing. – primeroot Mar 26 '21 at 16:52
  • I wouldn't think so. The standard problem, given $f(x,y) = ax^2 + bxy + c y^2,$ is to describe the prime values of $f.$ A related problem, given some $x^2 + n y^2,$ is to describe the squares with $\gcd(x,y) = 1,$ especially the prime squares. The best known book with background is Cox, Primes of the Form $x^2 + n y^2.$ I have the first edition, there is a second, it corrects at least one typo in a formula that I use. – Will Jagy Mar 26 '21 at 16:59
  • @DonaldSplutterwit Thanks for the link. The descriptions there are exactly what I'm thinking of. Perhaps other than "squares of primes" or "prime squares", there is no other name. – primeroot Mar 26 '21 at 17:00
  • @WillJagy I appreciate your comments, they have helped me resolve this question. – primeroot Mar 26 '21 at 17:04

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They are the numbers such that $k = p^2$, where $p$ is a prime, and also $1$.

Duncan Ramage
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