Is there a bibliometric that weighs citations based on their number?
If I understand you correctly, you are referring to a concept called citing-side normalization (see e.g. Waltman &amp; vanEck 2013), where the number of references of citing papers are used as weighting factors to account for the different lenghts of reference lists in different fields (citation density). This type of normalization is relatively common within bibliometrics, so I think that the other answers are incorrect in saying that they don't exist. One well-known example of such an index is the SNIP-indicator (see Waltman et al., 2013 and Leydesdorff &amp; Opthof, 2010), which is a journal-level indicator. Another example is the recently proposed article-level CSNCR-index (citation score normalized by cited references, see Bornmann &amp; Haunschild, 2016), but this uses the mean number of cited references in a field to normalize citations (so not references list length in individual papers).

If you look for references to "citing-side normalization" you will find other examples of similar approaches, at journal or article levels. I'm not sure if I've seen author-level indices based on this though (but averages of article-level indicies can of course be used). In general, there are a number of bibliometric indices that control for the different citations densities in different fields (not only using citing-side normalization). These are all based on some sort of field normalization, which can be done for field classes (in e.g. Scopus or Web of Science) or other article groupings (see e.g. the cluster-based normalization used at CWTS in Leiden).