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 21). When one digit is dropped from a "d"-digit integer, there are "d" integers in the sum and each is "d"-1 digits long. In general, when "n" digits are dropped from a "d"-digit integer, the number of integers in the sum is equal to "d"! / ("n"!("d" - "n")!), or the combination of "n" digits taken 2, 3, 4... at a time. For example, when "d" = 20 and "n" = 3, there are 20! / (3!(20 - 3)!) = 1,140 integers in the sum. In base ten, the integers31171093 01159390698 and 47989422298181591480943 are equal to their missing-digit sums when