Name of a quantity related to Shannon entropy, but with squared logarithm

Question

Let $k\geq 1$. For any probability distribution $p=(p_n)_{n=1}^\infty$ over $\mathbb{N}$, let $$ H_k(p) \stackrel{\rm def}{=} \sum_{n=1}^\infty p_n \log^k\frac{1}{p_n} \in[0,\infty] $$ For $k=1$, $H_1(p)=H(p)$ is the (Shannon) entropy. Does $H_2(p)$ have a name? Is it a standard quantity?