2005 AMC 10B Problem 11

The first term of a sequence is 2005. Each succeeding term is the sum of the cubes of the digits of the previous term. What is the {2005}^{\text{th}} term of the sequence?

\mathrm{(A)} 29 \qquad \mathrm{(B)} 55 \qquad \mathrm{(C)} 85 \qquad \mathrm{(D)} 133 \qquad \mathrm{(E)} 250

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.