Special Numbers

Thanks to Jeff Y. for presenting this puzzle to the Bohr Group.

A special number is a positive integer that is evenly divisible by all of its nonzero digits. Some special numbers are 7, 10000, 909, 16824, 84672, and, yes, even 30639636. The number 13 is not special. It is divisible by 1 but not 3. Zero is not special by definition.

Prove that there cannot be 14 consecutive special numbers, and find a list of 13 consecutive special numbers.