When the partial sum is visible immediately to the right of the summed digits, erase the said digits;
If after dropping some initial digits we are left with say 4.4.8…, the first 4 matches the second 4 without being technically a sum. We would drop the one digit and continue with 4.8… Correct? By "visible immediately to the right of the summed digits", you mean the *digits of the sum* are immediately to the right of the summed digits? So if after dropping some initial digits we are left with say 7.8.9.0.0.2.4.1…, we find a match after 5 (but not after 3) digits. Correct? Neither of the above situations (repeating-digit start, leading zeros) occurs in my search but being explicit is always helpful in creating an algorithm. For the number-of-dropped-digits sequence, I have seven terms: 2, 2, 10654, 61938, 59585723, 2, 195. The respective sums are: 4, 5, 47894, 278677, 268108667, 8, 933.