[seqfan] Re: Numbers that are divisible by the product of their digits
hv at crypt.org
hv at crypt.org
Sun Jun 18 10:48:38 CEST 2017
Charles Greathouse <charles.greathouse at case.edu> wrote:
:A007602 is the sequence of numbers that are divisible by the product of
:their digits. Does anyone know how this sequence grows?
Taking the simpler case of base 3, from 1 .. 3^n-1 the number of values
consisting of 1s and precisely k 2s is sum_k^n{ C(k, i) } = C(n+1, i+1).
Assuming that asymptotically such a number is divisible by its digit
product 1 / 2^k of the time, that gives me an expected count up to 3^n of:
sum_{i=0}^n { 2^(-i) C(n+1, i+1) }
Comparing that against actual counts gives:
n actual estimate 3^n
1 2 2.50 3
2 4 4.75 9
3 8 8.12 27
4 14 13.19 81
5 23 20.78 243
6 33 32.17 729
7 47 49.26 2187
8 80 74.89 6561
9 126 113.33 19683
10 169 171.00 59049
11 241 257.49 177147
12 412 387.24 531441
13 623 581.86 1594323
14 875 873.79 4782969
15 1303 1311.68 14348907
16 2031 1968.52 43046721
17 3017 2953.78 129140163
18 4417 4431.68 387420489
19 6631 6648.51 1162261467
20 10081 9973.77 3486784401
.. which looks pretty accurate, accepting that the "actual" results are
not evenly distributed (because eg numbers with a single 2 will be even
precisely when the length is odd).
I do not know if it is true or provable that this will asymptotically
approach the actual counts; but if it is, extending it to base 6 should
give an example usefully similar in structure to base 10 but of lesser
complexity.
Hugo
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