In mathematics, the persistence of a number is a term used to describe the number of times one must apply a given operation to an integer before reaching a fixed point, i.e. until further application does not change the number any more.
Usually, this refers to the additive or multiplicative persistence of an integer, which is how often one has to replace the number by the sum or product of its digits until one reaches a single digit. Because the numbers are broken down into their digits, the additive or multiplicative persistence depends on the radix. In the remaining article, we will assume a radix of 10.
The single-digit final state reached in the process of calculating an integers's additive persistence is its digital root. Put another way, a number's additive persistence is the measure of how many times we must sum the digits it takes us 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 number of multiplicative persistence 3.
Smallest numbers of a given persistence
For a radix of 10, there is thought to be no number with a multiplicative persistence > 11. The smallest numbers with persistence 0 through 11 are: 0, 10, 25, 39, 77, 679, 6788, 68889, 2677889, 26888999, 3778888999, 277777788888899 (by cleverly using the specific properties of numbers in this sequence, it can be calculated in a fraction of a second).
The additive persistence of a number, however, can become arbitrarily large (proof: For a given number n, the persistence of the number consisting of n repetitions of the digit 1 is 1 higher than that of n). The smallest numbers of additive persistence 0 through 4 are: 0, 10, 19, 199, 19999999999999999999999. The next number in the sequence (the smallest number of additive persistence 5) is 2 × 102×(1022 − 1)/9 − 1 (that is, 1 followed by 2222222222222222222222 9's). For any fixed base, the sum of the digits of a number is proportional to its logarithm; therefore, the additive persistence is proportional to the iterated logarithm.
* Sequence A003001 of smallest numbers for given multiplicative persistence in the On-Line Encyclopedia of Integer Sequences