What are "number_of_routing_shards"?

It's related to index splitting. For more details see the Elasticsearch Split Index docs. In order to allow for index splitting (on an existing shard), one must have this setting set at index creation time. Factorising the value of number_of_routing_shards / shards = a * b * c * ... gives you the potential split factors (a, b, c etc.) you can use in the lifetime of an index.