Editing Persistence of a number (section)

==Smallest numbers of a given additive persistence==
The additive persistence of a number, however, can become arbitrarily large ([[mathematical proof|proof]]: for a given number <math>n</math>, the persistence of the number consisting of <math>n</math> repetitions of the digit 1 is 1 higher than that of <math>n</math>). The smallest numbers of additive persistence 0, 1, 2, ... are:
:0, 10, 19, 199, 19999999999999999999999, ... {{OEIS|A006050}}
The next number in the sequence (the smallest number of additive persistence 5) is 2&nbsp;×&nbsp;10<sup>2×(10<sup>22</sup>&nbsp;−&nbsp;1)/9</sup>&nbsp;−&nbsp;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 [[tetration]]ally.