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
Perfect digit-to-digit invariant in all bases): When the convention formula_27 is used the following numbers are also PDDIs (as well as 0, in all bases): Perfect digit-to-digit invariant A perfect digit-to-digit invariant (PDDI) (also known as a Munchausen number) is a natural number that is equal to the sum of its digits each raised to the power of itself. In any base, there are only a finite number of Munchausen numbers. For example, in base 10 (decimal) there are only four: 0, 1, 3435 and 438579088. A number formula_1 in base formula_2 with digits formula_3 (where each formula_4 is an integer such that