Limited persistence (part 2)

I lied about leaving the other bases to you.

In base 2, as I said, every number > 1 has persistence 1. (Either it contains a 0, so goes to 0, or it’s all 1s, so goes to 1.)

In base 3, only one number in each decade (of the form 222…2223) is worth looking at. From 222223 on most of the digit products seem to contain zeros so the persistence is 2. 223 and 22223 also have persistence 2. The only numbers up to 1000 digits with larger persistence seem to be 2223 and 2222222222222223 both with persistence 3.

For bases 3 through 16, as far as I’ve checked in each:

BaseMax persistMin example
332223
4
33334
5633444444444444444444445
65244456
784445555555555556667
853335555778
9725777779
10
1127777778888889910
1112399999aaaaaaaaaaaaaaaaaaaaaaa11
127357777779912
13147777779aaaaaaaaabcccccc13
141355599999999999999aaaabbbbbb14
15112bbbbccccdddddde15
168379bdd16

One thing that seems to be happening is that you tend to get larger maximum persistences in prime bases, smaller ones in composite bases. Presumably that’s because of the analog to the 5-and-even situation in base 10: in a prime base, the digit product cannot be a multiple of the base, while in a composite base it can, in which case the digit product ends in a zero and terminates the sequence. Notice how every prime base from 5 on up has larger maximum persistence than the subsequent base. Also, recall 12 and 16 have respectively four and three proper divisors larger than 1, and then notice how small the maximum persistence is in bases 12 and 16 compared to bases 10, 11, 13, 14, and 15.

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