# Self-numbers and repunits

Self-numbers or Colombian numbers (A003052 in the OEIS) are natural numbers which are not the sum of a smaller number and the sum of its digits. Repunits (in base 10) are numbers consisting only of 1's.

• a) Are there infinitely many Self-numbers which are repunits?

• b) Are there infinitely many repunits which are not Self-numbers?

Are there infinitely many repunits which are not Self-numbers?

Yes. Consider any number with 1 repeated R_n times followed by n zeroes.
$$10, \; \underbrace{11111111111}_{11\;\text{times}}00, \;\underbrace{11\cdots11}_{111\;\text{times}}000, \; \cdots$$

Are there infinitely many Self-numbers which are repunits?

Yes. Referring to the following reduction test from this wiki (similar approach as this answer of mine):

We can work backwards and always get values for b so infinite such values exist.
Example - Start with 110. Add 1 and append a 1 at the beginning to get the next self number. There will be infinite repunits in the process.

• The "underbrace" mathjax looks broken in my browser (Chrome on MacOS): i.imgur.com/gkF1YgJ.png
– Bass
May 31, 2023 at 7:52
• @Bass no idea what is causing that issue, i will replace the mathjax with an image equivalent if i cant find a fix. May 31, 2023 at 15:26