Smallest Numbers of A Given 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 1050. The smallest numbers with persistence 0, 1, ... are:
- 0, 10, 25, 39, 77, 679, 6788, 68889, 2677889, 26888999, 3778888999, 277777788888899, ... (sequence A003001 in OEIS)
By cleverly using the specific properties of numbers in this sequence, the above terms 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, 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, ... (sequence A006050 in OEIS)
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.
Read more about this topic: Persistence Of A Number
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