Persistence of a number

In mathematics, the persistence of a number is the number of times one must apply a given operation to an integer before reaching a fixed point at which the operation no longer alters the number.

Usually, this involves additive or multiplicative persistence of a non-negative 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

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.

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

In base 10, there is thought to be no number with a multiplicative persistence > 11: this is known to be true for numbers up to 10^20,000.^[1][2] The smallest numbers with persistence 0, 1, 2, ... are:

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 in increasing order (with the exception of the second number, 10), 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.^[1]

Properties of additive persistence

• The additive persistence of a number is smaller than or equal to the number itself, with equality only when the number is zero.
• For base ${\displaystyle b}$ and natural numbers ${\displaystyle k}$ and ${\displaystyle n>9}$ the numbers ${\displaystyle n}$ and ${\displaystyle n\cdot b^{k}}$ have the same additive persistence.

More about the additive persistence of a number can be found here.

Smallest numbers of a given additive persistence

The additive persistence of a number, however, can become arbitrarily large (proof: for a given number ${\displaystyle n}$, the persistence of the number consisting of ${\displaystyle n}$ repetitions of the digit 1 is 1 higher than that of ${\displaystyle n}$). The smallest numbers of additive persistence 0, 1, 2, ... are:

0, 10, 19, 199, 19999999999999999999999, ... (sequence A006050 in the OEIS)

The next number in the sequence (the smallest number of additive persistence 5) is 2 × 10^{2×(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 its logarithm; therefore, the additive persistence is at most proportional to the iterated logarithm, and the smallest number of a given additive persistence grows tetrationally.

Functions 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 with or step to a single-digit number.

References

Literature

• Zbl 1058.11001
  .
• Meimaris, Antonios (2015). On the additive persistence of a number in base p. Preprint.