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

Terms

a(0) =42a(1) =281a(2) =375a(3) =416a(4) =593a(5) =736a(6) =907a(7) =975a(8) =1096a(9) =1193a(10) =1298a(11) =1330a(12) =1401a(13) =1420a(14) =1463a(15) =1577a(16) =1633a(17) =1707a(18) =2281a(19) =2288a(20) =2460a(21) =2712a(22) =2759a(23) =3085a(24) =3135a(25) =3412a(26) =3599a(27) =3722a(28) =3823a(29) =4328