  The partitions of the integers can be expressed exactly in an iterative and
closed-form expression. This equation is derived from distributing the
partitions of a number in a network that locates each partition in a unique and
orderly position. From this representation an iterative equation for the
function of the number of divisors was derivated. Also, the number of divisors
of a integer can be found from a new function called the trace of the number n,
trace(n). As a final preliminary result, using the Bressoud-Subbarao theorem,
we obtain a new iterative representation of the sum of divisor function. Using
this theorem it is possible to derive an iterative equation for any divisor
function and all their networks representation will exhibits a self-similarity
behavior. We must then conclude that the intricate nature of the divisor
functions results from the fractal nature of the partition function described
in the present work.
