Harold, Tanya, and Ulysses paint a very long picket fence. Harold starts with the first picket and paints every \(h\)th picket; Tanya starts with the second picket and paints every \(t\)th picket; and Ulysses starts with the third picket and paints every \(u\)th picket. Call the positive integer \(100h+10t+u\) \(\textit{paintable}\) 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.
(第二十届AIME1 2002 第9题)