I am surprised that Mathworld Wolfram did not list "Arithmetic Derivative" but Wiki has
You can use Arithmetic Derivative to formulate Goldbach's conjecture, twin prime and prime triples in an alternative way.
E. J. Barbeau, "Remark on an arithmetic derivative", Canad. Math. Bull. 4 (1961), 117 http://www.maa.org/mathland/mathtrek_03_22_04.html
Ufnarovski and Åhlander (2003) http://www.cs.uwaterloo.ca/journals/JIS/VOL6/Ufnarovski/ufnarovski.pdf
Why didn't Weisstein's Mathworld mention arithmetic derivative at all?