Persistence of a number assumed. The single-digit final state reached in the process of calculating an integer's additive persistence is its digital root. Put another way, a number's additive persistence counts how many times we must sum its digits to arrive at its digital root. The additive persistence of 2718 is 2: first we find that 2 + 7 + 1 + 8 = 18, and then that 1 + 8 = 9. The multiplicative persistence of 39 is 3, because it takes three steps to reduce 39 to a single digit: 39 → 27 → 14 → 4. Also, 39 is the smallest
Missing-digit sum dropping two, three and four digits, respectively. Here is the delete-2 sum, containing 12! / (2!(12 - 2)!) = 66 integers: In any base "b", there will be a set of delete-1 missing-digit sums with "b"+1 digits and first digits in the range 1..."b"-1, followed by "b" zeroes. Consider base-2 and base-3: Accordingly, the eleven-digit numbers 10,000,000,000 through 90,000,000,000 are trivial missing-digit sums in base ten, because their sums take this form: And in hexadecimal or base sixteen, the seventeen-digit numbers 10,000,000,000,000,000 through F0,000,000,000,000,000 are missing-digit sums of the same form. Searching for delete-1 missing-digit sums is simplified when one