1 s ▁( cons isting ▁of ▁the ▁number ▁written ▁as ▁▁ 1 s ). ▁Each ▁of ▁these ▁is ▁in ▁turn ▁equal ▁to ▁the ▁bin omial ▁coefficient ▁indexed ▁by ▁ ▁and ▁the ▁number ▁of ▁ 1 s ▁being ▁considered ▁( for ▁example , ▁there ▁are ▁ 1 0 - choose - 3 ▁binary ▁numbers ▁with ▁ten ▁digits ▁that ▁include ▁exactly ▁three ▁ 1 s ). ▁ ▁Currently , ▁powers ▁of ▁two ▁are ▁the ▁only ▁known ▁almost ▁perfect ▁numbers . ▁ ▁The ▁number ▁of ▁vertices ▁of ▁an ▁- dimensional ▁hyper c ube ▁is ▁. ▁Similarly , ▁the ▁number ▁of ▁- faces ▁of ▁an ▁- dimensional ▁cross - poly to pe ▁is ▁also ▁ ▁and ▁the ▁formula ▁for ▁the ▁number ▁of ▁- faces ▁an ▁- dimensional ▁cross - poly to pe ▁has ▁is ▁▁ ▁The ▁sum ▁of ▁the ▁reci pro cal s ▁of ▁the ▁powers ▁of ▁two ▁is ▁ 1 . ▁The ▁sum ▁of ▁the ▁reci pro cal s ▁of ▁the ▁squ ared ▁powers ▁of ▁two ▁is ▁ 1 / 3 . ▁ ▁The ▁smallest ▁natural ▁power ▁of ▁two ▁whose ▁decimal ▁representation ▁begins ▁with ▁ 7 ▁is ▁ ▁See ▁also ▁▁ 2 0 4 8 ▁( video ▁game ) ▁ ▁Binary ▁number ▁ ▁Ge ometric ▁pro gression ▁ ▁G ould ' s ▁sequence ▁ ▁In ch w orm ▁Song ▁ ▁Integer ▁binary ▁log arith m ▁ ▁Oct ave ▁( elect ron ics ) ▁ ▁Power ▁of ▁ 1 0 ▁ ▁Power ▁of ▁three ▁▁ ▁Sum - free ▁sequence ▁ ▁References ▁ ▁Category : Binary ▁arithmetic ▁Category