examples of self numbers in a few selected bases

In base 10, the first few self numbers are 1, 3, 5, 7, 9, 20, 31, 42, 53, 64, 75, 86, 97 (listed in Sloane’s A003052). The recurrence relation $S_i=8\cdot {10}^{i-1}+S_{i-1}+8$ with $S_1=9$ gives the list 9, 97, 905, 8913…

In binary, the first few self numbers are 1, 4, 6, 13, 15, 18, 21, 23, 30, 32, 37, 39, 46, 48, 51, 54 (listed in A010061 of Sloane’s OEIS). The recurrence relation $S_i=2^k+S_{i-1}+1$ with $S_1=1$ and $k$ being the number of bits in the number $S_{i-1}$ has, gives the list 1, 4, 13, 30…

In hexadecimal, the first few self numbers are 1, 3, 5, 7, 9, 11, 13, 15, 32, 49, 66, 83, 100, 117, 134 (not currently listed in the OEIS). The recurrence relation $S_i=14\cdot {16}^{i-1}+S_{i-1}+14$ gives the list 1, 239, 3837, 61195…

In factorial base, the first few self numbers are 1, 4, 11, 18, 36, 43, 61, 68, 86, 93, 111, 118, 125, 132. I don’t know of a recurrence relation that will generate factorial base self numbers.

Lastly, in Roman numerals, the first few self numbers are 1, 3, 5, 7, 8, 9, 11, 13, 15.