# (no calc) Computing $a_{111}$ if $a_1 < a_2 < a_3 < …$ lists all integers that can be expressed as the sum of distinct repunits

Recently at a math competition, our team was given the problem: "A repunit is a positive integer, all of whose digits are 1's. Let $a_1 < a_2 < a_3 < ...$ be the list of all positive integers that can be expressed as the sum of distinct repunits. Compute $a_{111}$"

This problem was at a high school math competition. My skill level ends around Algebra II, as I am taking the course now in school. To be honest, I'm lost as to what $a_1 < a_2 < a_3 < ...$ represents when it says it is the "list of all the positive integers that can be expressed as the sum of distinct repunits".

I would give work, to begin with, I am utterly lost. I'm sorry, but anything helps.

• Welcome to stackexchange. Hint: the sequence begins $1, 11, 12, 111, 112, 122, 123$ since $1$, $11$ and $111$ are the first three repunits. The next one is $1111$. That should get you started. – Ethan Bolker Nov 12 '17 at 19:49
• Thanks for the welcome. This has gotten me onto the right track. I've been doing some more of the sequence. Would $a_{12}$ be 11111? – sunnysmog Nov 12 '17 at 20:00
• Yes, I think so. – Ethan Bolker Nov 12 '17 at 21:48