The Dedekind psi function is defined for a positive integer $m>1$ as

$$\psi(m)=m\prod_{\substack{p\mid m\\p\text{ prime}}}\left(1+\frac{1}{p}\right)\tag{1}$$ with the definition $\psi(1)=1$. See the Wikipedia Dedekind psi function (I add also the reference [1]). I wondered (while I was studying elementary problems similar than some problems that are in the literature of the Euler's totient function) if the following advanced problem is in the literature.

Problem. Given an integer $X\geq 1$ we define the set $$\{\text{integers } 1\leq n\leq X:\text{ the equation }\psi(x)=n\text{ has a unique solution } x\}\tag{2}$$ and we denote the cardinality of this set as $$f(X):=\#\{ 1\leq n\leq X:\psi(x)=n\text{ has a unique integer solution } x\geq 1\}.$$ Determine what is the asymptotic or how grows $f(X)$ as $X\to\infty$.

Question. What work can be done to get a not obvious statement $$f(X)=\text{main term}+\text{error term}\tag{3}$$ as $X\to\infty$? I'm asking a statement at research level if isn't in the literature*, you can provide your statement about this asymptotic behaviour $(3)$ of $f(X)$ using Landau notation or other notation. Many thanks.

*I don't know if this problem is in the literature, if this problem is in the literature answer my question as a reference request and I try to search and read the statement from the literature.

A related sequence is A203444 from the OEIS, but I believe that the sequence of $n$'s such that $\psi(x)=n$ has a unique solution $x$, isn't in the OEIS.

I've calculated with a Pari/GP script the first positive integers $n$ such that $\psi(x)=n$ has a unique solution, its solution $x$. For very small integers $X$ it seems that $f(X)\ll X$ in Vinogradov notation (let's say $f(X)\approx \frac{X}{10}$).


[1] Tom M. Apostol, Introduction to analytic number theory, Undergraduate Texts in Mathematics, New York-Heidelberg: Springer-Verlag (1976).

  • $\begingroup$ As was said if this problem is in the literature feel free to refer that this is a well known problem. I suspect that this kind of problems should be difficult since other similar that are in the literature for the Euler's totient function were very difficult. Any case I'm asking about what work can be done to get a statement $(3)$ at research level, for very large $X$ (if you can deduce the corresponding proposition for $X\to\infty$ it is better, of course). $\endgroup$ – user142929 Apr 29 at 18:12
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    $\begingroup$ Dedekind psi tabulated at oeis.org/A001615 – I think the sequence under discussion is very like oeis.org/A291109 $\endgroup$ – Gerry Myerson Apr 30 at 4:25
  • $\begingroup$ Now I'm confused, maybe the problem is well known, if the Problem in the body of the post is known refer the literature in comments or answering the Question as a reference request. Many thanks @GerryMyerson $\endgroup$ – user142929 Apr 30 at 4:40
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    $\begingroup$ Sorry, I'm the one who is confused. 291109 resembles numbers not in the range of psi, rather than numbers appearing exactly once as values of psi. $\endgroup$ – Gerry Myerson Apr 30 at 5:07

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