I wouldn’t know the right list.
Here is a list, along with their respective ages upon taking office for your enjoyment. Polk is easily amongst the most effective presidents of all time and didn't seek re-election, although he certainly didn't help alleviate the sectional crisis any more than the rest of the one-term cans who ran the country from 1837 up to Lincoln.
1789-1797: George Washington (57)
1797-1801: John Adams (61)
1801-1809: Thomas Jefferson (57)
1809-1817: James Madison (57)
1817-1825: James Monroe (58)
1825-1829: John Quincy Adams (57)
1829-1837: Andrew Jackson (61)
1837-1841: Martin Van Buren (54)
1841-1841: William Henry Harrison (68)
1841-1845: John Tyler (51)
1845-1849: James Knox Polk (49)
1849-1850: Zachary Taylor (64)
1850-1853: Millard Fillmore (50)
1853-1857: Franklin Pearce (48)
1857-1861: James Buchanan (65)
1861-1865: Abraham Lincoln (52)
1865-1869: Andrew Johnson (56)
1869-1877: Ulysses S. Grant (46)
1877-1881: Rutherford Hayes (54)
1881-1881: James Garfield (49)
1881-1885: Chester Arthur (51)
1885-1889: Grover Cleveland (47)
1889-1893: Benjamin Harrison (55)
1893-1897: Grover Cleveland (55)
1897-1901: William McKinley (54)
1901-1909: Theodore Roosevelt (42)
1909-1913: William Howard Taft (51)
1913-1921: Woodrow Wilson (56)
1921-1923: Warren G. Harding (55)
1923-1929: Calvin Coolidge (51)
1929-1933: Herbert Hoover (54)
1933-1945: Franklin D. Roosevelt (51)
1945-1953: Harry Truman (60)
1953-1961: Dwight Eisenhower (62)
1961-1963: John F. Kennedy (43)
1963-1969: Lyndon B. Johnson (55)
1969-1974: Richard Nixon (56)
1974-1977: Gerald Ford (61)
1977-1981: Jimmy Carter (52)
1981-1989: Ronald Reagan (69)
1989-1993: George H.W. Bush (64)
1993-2001: Bill Clinton (46)
2001-2009: George W. Bush (54)
2009-2017: Barack Obama (47)
2017-2021: Donald J. Trump (70)
2021-2025: Joe Biden (78)
2025-2029: Donald J. Trump (78)