In a 2010 paper, Chao, Chui and Jost describe a family of metrics based on Hill numbers with a single parameter (q) that determines how abundance is weighted. I call these Jost indexes of order q. The numerical value of the metric for any q is an effective number of species.
The parameter q is the power to which the species frequencies are raised. Higher values of q thus give lower weights to low-abundance species. With q=0 the Jost index is exactly equivalent to richness. With q=1, the Jost index is the effective number of species given by the Shannon entropy. The q parameter can take real values, not just integers, which allows intermediate metrics with different weightings of abundance.
Chao, Chui and Jost (2010)