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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?

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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):
enter image description here

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.

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

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