The '''arithmetic derivative''' of an integer (more specifically, the
'''Lagarias arithmetic derivative''') is a function defined for integers, based on prime
factorization, by analogy with the product rule for the derivative of a function that is
used in mathematical analysis. Accordingly, for natural numbers n, the arithmetic 
derivative D(n) is defined as follows:

;*D(0) = D(1) = 0.
;*D(p) = 1 for any prime p.
;*D(mn) = D(m)n + mD(n) for any m,n ∈ N. (Leibniz rule for derivatives).

Additionally, for negative integers the arithmetic derivative may be defined as -D(-n) (n < 0).

; Examples

D(2) = 1 and D(3) = 1 (both are prime) so if mn = 2 * 3, D(6) = (1)(3) + (1)(2) = 5.

D(9) = D(3)(3) + D(3)(3) = 6

D(27) = D(3)*9 + D(9)*3 = 9 + 18 = 27

D(30) = D(5)(6) + D(6)(5) = 6 + 5 * 5 = 31.

; Task

Find and show the arithmetic derivatives for -99 through 100.

; Stretch task

Find (the arithmetic derivative of 10^m) then divided by 7, where m is from 1 to 20.

; See also

;* [[oeis:A003415|OEIS:A003415 - a(n) = n' = arithmetic derivative of n.]]
;*[[wp:Arithmetic_derivative|Wikipedia: Arithmetic Derivative]]