There are a few Presidents that did not have a Vice President.
John Tyler / 1841 - 1845 - no Vice President
Millard Fillmore / 1850 - 1853 - no Vice President
Andrew Johnson / 1865 - 1869 - no Vice President
Chester A. Arthur / 1881 - 1885 - no Vice President
There are Presidents that have gone their terms without a Vice President, but listed above are the ones that never had one.