# Find the least integer $k$ such that $\left[H_k\right]=n$ for a given positive integer $n$

At the moment I am curious about harmonic numbers at their properties and came across the following question.

We are given $$n\in\mathbb{N}$$. We are to find the least integer $$k$$ such that $$\left[H_k\right]=n$$ where $$H_n$$ is the $$n$$-th harmonic number and $$[\cdot]$$ is the integer part.

My attempt. It is known that $$H_n=\gamma+\psi(n+1)$$ where $$\psi(x)$$ is digamma function. So I got $$\left[H_k\right]=n\\n\leq H_k

where $$\psi^{(-1)}(x)$$ is inverse function of $$\psi(x)$$. $$\psi(x)$$ is increasing for $$x>0$$.

And I stuck here. I do not know how to show that there exists integer $$k$$ in interval $$\left[\psi^{(-1)}(n-\gamma)-1;\psi^{(-1)}(n-\gamma)-1\right)$$ at all.

I suppose that I've chosen too complicated way to tackle this problem. I appreciate any hint or answer. Thanks!