Persistence of a number
Usually, this involves 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 remainder of this article, base ten is 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 number of multiplicative persistence 3.
Smallest numbers of a given persistence
- 0, 10, 25, 39, 77, 679, 6788, 68889, 2677889, 26888999, 3778888999, 277777788888899. (sequence A003001 in the OEIS)
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 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 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 200 digits if they do exist.
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:
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. More about the additive persistence of a number can be found here.