Does a divorce take effect if the husband says to his wife, "By Allah, if you do this, you are divorced," and then claims he intended only to prevent her, knowing that the wife did the thing and six months have passed since then?

Since you did the thing your husband swore by divorce upon, due to forgetfulness, the preponderant opinion is that your husband did not break his oath, and no divorce has occurred. It is stated in Rawdat al-Talibin: "If he intended to prevent her from disobedience, but she forgot, Al-Ghazali said: She is definitely not divorced, due to the absence of disobedience." Therefore, you are still under your husband's marital bond, and his intimacy with you is permissible. There is no sin or expiation upon you or him.