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 $$\left[H_k\right]=n\\n\leq H_k<n+1\\n\leq \gamma+\psi(k+1)<n+1\\n-\gamma\leq \psi(k+1)<n-\gamma+1\\\psi^{(-1)}(n-\gamma)-1\leq k<\psi^{(-1)}(n-\gamma+1)-1 $$

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!


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