September 2019 - Solution
Many of you got the 1, 2, 4, 8, 62 solution, but there are others. The closest to the original set of notes is 1, 5, 10, 34, 50 (replacing the $20 bill with a $34 note).
Thanks to JJ Rabeyrin who told us about the relevant paper "What This Country Needs is an 18-cent Piece".
Thanks to Yasodhar (Pattu) Patnaik, for sending us a nice visualization you can play with: https://yamapa2.github.io/
Thanks also to Collin Pearce for a solution to a different question (to find a range of values, such that the standard set of bills, will give a 4% chance of selecting two different values with the same bill set).