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 $f(X)\approx \frac{X}{10}$).

## References:

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

Problemin the body of the post is known refer the literature in comments or answering theQuestionas a reference request. Many thanks @GerryMyerson $\endgroup$ – user142929 Apr 30 at 4:40