Governing body, the FIA, gave drivers a choice of numbers between 2 and 99 with the number one reserved for the defending world champion, currently Germany's Red Bull driver Sebastian Vettel.
Vettel will opt for the number five in 2015 should he lose the world title he has held since 2010.
The decision to add permanent numbers to the vehicles was taken to boost the sport's marketing possibilities.
"2 little ducks! #22 #jenson22 my new race number guys, great memories," he wrote.
Button had the number 22 on his Brawn F1 car when he won his only world title in 2009.
His new teammate, Kevin Magnusson, will carry number 20, the same as he used in last year's World Series by Renault championship, one of motor-racing's second tier competitions.
"My driver number will be 20, which I won the @WSR_Live3.5 last year," tweeted the Dane.
Mercedes teammate Nico Rosberg opted for six and tweeted: "Cool. Got number 6 for my remaining F1 career! My future wife's and my dad's lucky number.
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