A recreational math problem, dubbed "Insert and Add", asks: What is the least integer m that requires no less than n insertions of plus signs so that, after performing the addition(s), we arrive at a single digit? (See the last page here: http://orion.math.iastate.edu/butler/papers/16_03_insert_and_add.pdf)
It is similar to finding the additive persistence of n, but instead of merely counting the number of digital sums required to arrive at a single digit it counts the minimum number of plus signs inserted during that process.
10 is the smallest number that requires one plus sign: 1+0=1. 19 is the smallest to require two: 1+9=10 -> 1+0=1. 118 is the smallest to require three: 1+1+8=10 -> 1+0=1; alternatively we can try 1+18=19 -> 1+9=10 -> 1+0=1; and finally we can try 11+8=19 -> 1+9=10 -> 1+0=1.
3187, and 3014173 are the next two numbers in the sequence.
Now observe that all of these numbers (10, 19, 118, 3187, 3014173) have a digital root of 1.
Is it obvious that all future terms in this sequence will have digital root 1?
The sequence is https://oeis.org/A293929.