Every tweet that contains @POTUS (or the name of any other elected official) is being archived by the government, or, as Gawker puts it:

The White House Is Archiving Every Tweet Begging @POTUS for Sex

On Twitter, the White House automatically archives “tweets” from official White House accounts, “direct messages” sent to or from official White House accounts, and “mentions” (tweets from other users to official White House accounts; these tweets contain an @ and the username of an official White House account (e.g., @WhiteHouse).