The October record for home runs was once four in the National League (creating a three league tie) until Sexson & Sosa set new records in 2001. Before them the record holders were Dave Parker (1985), Mike Schmidt (1980) & Ned Williamson (1884).
Do you know how many of the home runs in a month record holders are members of the 500 Home Runs Club? Check the page and see for yourself.
No comprehensive weekly home run records yet, but the Major League record is ten set by Frank Howard of the Washington Senators (AL) between May 12th through the 18th of 1968. The next closest player to that mark is Shawn Green who had nine for the Los Angeles Dodgers (NL) between May 19, 2002 and May 25, 2002.