On the first of the month in June of 1868 James Buchanan, the only bachelor to be elected until Grover Cleveland in 1884, died and, interesting, was the last Democratic candidate elected until Cleveland's '84 win. The fifth of June belongs to Ronald Reagan in 2004. June eighth from 1845 is when Andrew Jackson succumbed to old age. Four years after Old Hickory went, Young Hickory, James K. Polk had the shortest retirement from office at such a young age dying June fifteenth, 1849, just three months after leaving a self-imposed single term. Next deathday in the month of June is the twenty-fourth in 1908 and the presidency...also number twenty-two, but only in the presidency, Cleveland the popular vote winner in '84, '88 and '92, but only in the Electoral College in '84 and '92. The twenty-eighth of the month in 1836 sees the death of James Madison.
That's SIX days and SIX presidential deaths.
June is just the second most interesting, if not morbid, and pointless, presidential deathday related pfact!