A052058 Largest palindromic substring (without leading zeros) of 2^n is a repdigit of minimum length 2.

16, 24, 25, 26, 27, 38, 39, 40, 42, 43, 44, 45, 46, 51, 95, 96, 108, 169, 191, 193, 198, 202, 205, 247, 250, 316, 317, 379, 386, 421, 422, 423, 425, 485, 486, 488, 589, 592, 642, 643, 659, 731, 736, 758, 759, 800, 835, 839, 927, 971, 972, 978