A Sequence of Summed Cubes

Take
a certain 3-digit integer and sum the cubes of its digits to form a new
integer. Do this again to get back to the original integer. What is the
starting integer? Find all possible solutions. Example:
217 gives 2
^{3}+1^{3}+7^{3} =
352. 352 gives 3^{3}+5^{3}+2^{3} =
217. |

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Last Updated: January 13th, 2010.