 Halo spacing from a differential mass function

For comoving differential number density $n(M)=dn/dM$, the abundance in an order-unity logarithmic mass bin is approximately $Mn(M)$. A characteristic mean spacing is therefore $[Mn(M){]}^{-1/3}$. A cumulative mass fraction cannot replace this number directly; cumulative halo counts require integrating $n(M)$ over mass.