Conjecturally, a power of 2 written in base 3 cannot have this many 0's.

Terms

a(0) =115a(1) =124a(2) =139a(3) =243a(4) =367a(5) =445a(6) =783a(7) =914a(8) =958a(9) =1095a(10) =1112a(11) =1200a(12) =1239a(13) =1312a(14) =1487a(15) =1752a(16) =1902a(17) =2013a(18) =2504a(19) =2583a(20) =2620a(21) =2697a(22) =2725a(23) =2754a(24) =2881a(25) =3015a(26) =3365a(27) =3443a(28) =3612a(29) =3673