For integers $n\geq 1$ with $$\operatorname{rad}(n)=\prod_{\substack{p\mid n\\p\text{ prime}}}p$$ we denote the squarefree kernel or radical of an integer $n$ (see if you want this Wikipedia). And $\varphi(n)$ denotes the Euler's totient funciton. Then while I was stuying the equation $$\operatorname{rad}(\varphi(2\cdot n))=2$$ in the context of constructible polygons with compass and straightedge, see this section of this Wikipedia, I wondered next variation $$\operatorname{rad}(\varphi(3\cdot n))=2\cdot 3.\tag{1}$$

Ater I search the resulting sequence of solutions of our equation $(1)$, the first few terms are $3,6,7,9,12,13,14,\ldots$, in The On-Line Encyclopedia of Integer Sequences, I would like to do a comparison to the sequence *A135412*. I know that there are terms in *A135412*, but don't satisfy our equation (1). You can read the definition of Heronian mean from this Wikipedia.

Question.I would like to know* if it is possible to prove or refute that:

If$n$satisfies$$\operatorname{rad}(\varphi(3 n))=6$$then our solution$n$equals three times the Heronian mean of two positive integers.

Many thanks.

I think that this question is curious. If a full answer isn't feasible I'm interested to know what work can be done.

*Additionally if you want to add remarks in your answer about different equations involving the Euler's totient function and/or the squarefree kernel, showing some remarkable relationship to the sequence of the Heronian means in some way feel free to provide me such feedback.