70

A "weird number" is a number that is less than the sum of its proper factor, yet is not equal to the sum of any subset of them.

70 is the smallest weird number. 

Its proper factors are 1, 2, 5, 7, 10, 14, and 35.  The sum of these is 74.  But you can not add up a subset of these numbers to get 70.

(Amy)