Who was the first US president to assume the office after the death of his predecessor?

In 1841, William Henry Harrison was in office for barely a month before passing away placing his vice president John Tyler as the new president. Harrison had given a long-winded inauguration speech from which he developed pneumonia and never recovered. So far he has been the shortest-serving president in American history. Tyler's surprise quick rise to the presidency was given the nickname "His Accidency."

https://www.whitehouse.gov/about-the-white-house/presidents/john-tyler/