Why did some presidents not have vice presidents?

Until 1967 when the 25th amendment was added to the Constitution, a vacancy in the office of vice-president was not filled until the next presidential election.
Consequently, if the president died and the vice president became president or the vice-president died in office, there was no vice-president until the next presidential election. Those presidents, such as Tyler, Fillmore, Andrew Johnson and Arthur , who were never elected as president, therefore never had a vice-president.