Examples
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 multiplicative persistence
For a radix of 10, there is thought to be no number with a multiplicative persistence > 11: this is known to be true for numbers up to 1020000. The smallest numbers with persistence 0, 1, ... are: :0, 10, 25, 39, 77, 679, 6788, 68889, 2677889, 26888999, 3778888999, 277777788888899. The search for these numbers can be sped up by using additional properties of the decimal digits of these record-breaking numbers. These digits must be sorted, and, except for the first two digits, all digits must be 7, 8, or 9. There are also additional restrictions on the first two digits. Based on these restrictions, the number of candidates for ''n''-digit numbers with record-breaking persistence is only proportional to the square of ''n'', a tiny fraction of all possible ''n''-digit numbers. However, any number that is missing from the sequence above would have multiplicative persistence > 11; such numbers are believed not to exist, and would need to have over 20,000 digits if they do exist.Properties of additive persistence
* The additive persistence of a number is smaller or equal to the number itself, with equality only when the number is zero. * For base and natural numbers and the numbers and have the same additive persistence. More about the additive persistence of a number can be founSmallest numbers of a given additive persistence
The additive persistence of a number, however, can become arbitrarily large (proof: For a given number , the persistence of the number consisting of repetitions of the digit 1 is 1 higher than that of ). The smallest numbers of additive persistence 0, 1, ... 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 at most proportional to itsFunctions with limited persistence
Some functions only allow persistence up to a certain degree. For example, the function which takes the minimal digit only allows for persistence 0 or 1, as you either start or step to a single-digit number.References
Literature
* * {{Classes of natural numbers Number theory