Let $n\geq 1$ an integer, in this post I denote the greatest prime dividing $n$ as $\operatorname{gpf}(n)$, and the product of the distinct prime numbers dividing $n$ as $$\operatorname{rad}(n)=\prod_{\substack{p\mid n\\p\text{ prime}}}p.$$ See it you want the Wikipedia Radical of an integer and the corresponding article from MathWorld Greatest Prime Factor.
*I'm interested in create interesting inequalities involving previous arithmetic functions, with a good mathematical content. After some experiments with these arithmetic functions I wondered next question.
Question. Are there infinitely many integers $n\geq 1$ such that $$\sqrt{\frac{\operatorname{rad}(n)}{2}}>\operatorname{gpf}(n)\tag{1}$$ holds? Many thanks.
The sequence of the first few integers satisfying $(1)$ starts as
$$105,210,330,385,390,420,429,455,462,\ldots$$
and I suspect that should be infinitely many of them. I don't know how to solve it, maybe using inequalities about different means, and invoking some theorem about prime numbers.
Appendix:
*I did more experiments combining these arithmetic functions, similar than previous or for example this different inequality for integers $n>1$
$$\frac{\sqrt{\operatorname{rad}(n)}}{\log n}>\operatorname{gpf}(n) ,$$
the first few solutions of this last inequality are $2,30030,34034,36465,39270,40755\ldots$
Thus the corresponding inequalities $<$ to $(1)$ and this last inequality seem that are satisfied for many much positive integers.
