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.

Title	examples of self numbers in a few selected bases
Canonical name	ExamplesOfSelfNumbersInAFewSelectedBases
Date of creation	2013-03-22 15:56:18
Last modified on	2013-03-22 15:56:18
Owner	PrimeFan (13766)
Last modified by	PrimeFan (13766)
Numerical id	5
Author	PrimeFan (13766)
Entry type	Example
Classification	msc 11A63