I want to find out the number of integers whose biggest non-trivial divisor is exactly $k$ times the smallest non-trivial divisor of that integer.

My thoughts are, that the smallest divisor $n$ has to be prime, otherwise the divisor would have smaller divisors, which would divide the larger integer too. Also the biggest divisor would be $kn$. Also I believe that the integers have to be in the form $mkn$, and then I would have to exclude some of these integers because the property does not hold, but i am not sure how to prove that.

  • 2
    $\begingroup$ Note that every number (with at least 1 non-trivial divisor) is the product of their smallest and largest non-trivial divisors. This means your number is $kn^2$ for $n$ prime and $k$ having no prime factors less than $n$. $\endgroup$ – Don Thousand Apr 2 at 0:46
  • $\begingroup$ so the smallest non-trivial divisor has to be prime, and then I get the integer $kn^2$, but the relationship to the number of such integers still eludes me $\endgroup$ – M-S-R Apr 2 at 0:59

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