Which president was the only vice president to become president without following the president for whom he served as vice-president?

Of the 14 U. S. Vice Presidents who became President, the only one whose presidency did not immediately follow his vice presidency is Richard M. Nixon. He did run for president while still the incumbent vice president, but he lost that election (1960) and did not run again for president until 8 years later.

It's interesting that Nixon preceded Lyndon Johnson as vice president, but Johnson preceded Nixon as president.