[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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