# Persist List

Multiplicative persistence is a complex term but a simple concept. Take a number, multiply its digits, repeat. Sooner or later the result is a single digit:

25 → 2 x 5 = 10 → 1 x 0 = 0 (mp=2)
39 → 3 x 9 = 27 → 2 x 7 = 14 → 1 x 4 = 4 (mp=3)

So 25 has a multiplicative persistence of 2 and 39 a multiplicative persistence of 3. Each is the smallest number with that m.p. in base-10. Further records are set by these numbers:

77 → 49 → 36 → 18 → 8 (mp=4)
679 → 378 → 168 → 48 → 32 → 6 (mp=5)
6788 → 2688 → 768 → 336 → 54 → 20 → 0 (mp=6)
68889 → 27648 → 2688 → 768 → 336 → 54 → 20 → 0 (mp=7)
2677889 → 338688 → 27648 → 2688 → 768 → 336 → 54 → 20 → 0 (mp=8)
26888999 → 4478976 → 338688 → 27648 → 2688 → 768 → 336 → 54 → 20 → 0 (mp=9)
3778888999 → 438939648 → 4478976 → 338688 → 27648 → 2688 → 768 → 336 → 54 → 20 → 0 (mp=10)

Now here’s base-9:

25[b=9] → 11 → 1 (mp=2)
38[b=9] → 26 → 13 → 3 (mp=3)
57[b=9] → 38 → 26 → 13 → 3 (mp=4)
477[b=9] → 237 → 46 → 26 → 13 → 3 (mp=5)
45788[b=9] → 13255 → 176 → 46 → 26 → 13 → 3 (mp=6)
2577777[b=9] → 275484 → 13255 → 176 → 46 → 26 → 13 → 3 (mp=7)

And base-11:

26[b=11] → 11 → 1 (mp=2)
3A[b=11] → 28 → 15 → 5 (mp=3)
69[b=11] → 4A → 37 → 1A → A (=10b=10) (mp=4)
269[b=11] → 99 → 74 → 26 → 11 → 1 (mp=5)
3579[b=11] → 78A → 46A → 1A9 → 82 → 15 → 5 (mp=6)
26778[b=11] → 3597 → 78A → 46A → 1A9 → 82 → 15 → 5 (mp=7)
47788A[b=11] → 86277 → 3597 → 78A → 46A → 1A9 → 82 → 15 → 5 (mp=8)
67899AAA[b=11] → 143A9869 → 299596 → 2A954 → 2783 → 286 → 88 → 59 → 41 → 4 (mp=9)
77777889999[b=11] → 2AA174996A → 143A9869 → 299596 → 2A954 → 2783 → 286 → 88 → 59 → 41 → 4 (mp=10)

I was also interested in the narcissism of multiplicative persistence. That is, are any numbers equal to the sum of the numbers created while calculating their multiplicative persistence? Yes:

86 = (8 x 6 = 48) + (4 x 8 = 32) + (3 x 2 = 6)

I haven’t found any more in base-10 (apart from the trivial 0 to 9) and can’t prove that this is the only one. Base-9 offers this:

78[b=9] = 62 + 13 + 3

I can’t find any at all in base-11, but here are base-12 and base-27:

57[b=12] = 2B + 1A + A
A8[b=12] = 68 + 40 + 0

4[b=27] = 3B + 16 + 6
7[b=27] = 66 + 19 + 9
A[b=27] = 6 + 40 + 0
[b=27] = 3 + 2F + 13 + 3
[b=27] = 8 + 583 + 4C + 1 + 

But the richest base I’ve found so far is base-108, with fourteen narcissistic multiplicative-persistence sums:

4[b=108] = 3 + 1 + 
5[b=108] = 2 + 1 + 
7[b=108] = 6 + 1 + 
A[b=108] = 6 + 40 + 0
[b=108] = E + 3 + 1 + 
[b=108] = C + 2 + 1C + C
[b=108] =  + 6 + 30 + 0
[b=108] =  + F + 2 + 10 + 0
[b=108] =  +  + 4 + 3 + 1 + 
[b=108] =  +  + 6 + 5 + 3 + 10 + 0
[b=108] =  + C + 2 + 1 + 
[b=108] = 9 + 8 + 1 + 
5[b=108] = 3 + 280 + 0
8[b=108] = 7 + 1 + D + 8 + 5 + 1 + 

Update (10/ii/14): The best now is base-180 with eighteen multiplicative-persistence sums.

5[b=180] = 2 + 1 + 
7[b=180] = 4 + 2 + 
7[b=180] = 6 + 1 + 
8[b=180] = 4 + 3 + 
A[b=180] = 6 + 40 + 0 (s=5)
[b=180] = E + 3 + 1 + 
[b=180] = C + 7 + 1 + 
[b=180] =  + 2 + 1 + 
[b=180] =  + E8 + 
[b=180] =  + 6 + 30 + 0 (s=10)
[b=180] =  + A + 7 + 2 + 
[b=180] =  + C + 2 + 1 + 
[b=180] =  +  + 4 + 3 + 1 + 
[b=180] =  +  + 6 + 5 + 3 + 10 + 0
[b=180] = F + 8 + 1 +  (s=15)
[b=180] =  +  + C + 9 + 5 + 20 + 0
5[b=180] = 3 + 20 + 0
E[b=180] = 8 + 4 +  +  + 0 + 0

## One thought on “Persist List”

1. Norman Foreman, B.A. says:

You got the base-108 wrong. Doh!

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