Harold, Tanya, and Ulysses paint a very long picket fence.

Harold starts with the first picket and paints every h^{\text {th }} picket;

Tanya starts with the second picket and paints every t^{\text {th }} picket; and Ulysses starts with the third picket and paints every u^{\text {th }} picket.

Call the positive integer 100 h+10 t+u paintable when the triple (h, t, u) of positive integers results in every picket being painted exactly once. Find the sum of all the paintable integers.