Product Sum Ratios

I'm surprised I've not made a post about this topic before. The topic concerns the result when the product of the digits of a number is divided by the sum of its digits. Of interest is:

• when is the result of this division an integer?
• what numbers are associated with records in the size of this integer?

Clearly 1 will start the ball rolling with a record of 1 but after that the next number is 36 with a product of 18 and a sum to 9, giving a record of 2. 66 is next with a product of 36 and a sum of 12, giving a record of 3. Figure 1 shows a list of the record numbers up 10,000 followed by a plot of these numbers (Figure 2). 

Figure 1: permalink

Figure 2: permalink

Extending out to 100,000, Figure 3 shows an extended table and Figure 4 shows a plot of the full range of record-breaking numbers.

Figure 3: permalink

Figure 4: permalink

I won't show a table of results up to one million but here are the 59 record-breaking numbers (permalink):

1, 36, 66, 88, 257, 268, 279, 369, 459, 578, 579, 678, 789, 999, 2589, 2688, 2799, 3699, 3789, 4599, 4689, 4789, 5788, 5889, 7889, 8888, 18999, 25889, 26789, 26888, 27788, 28899, 37899, 38889, 45999, 46899, 47799, 47889, 55899, 56889, 57789, 58999, 78999, 257899, 258889, 267999, 277899, 278889, 367899, 377889, 378888, 457899, 459999, 489999, 588999, 678999, 688899, 778899, 778999

The results for the final number 778999 are:$$ \frac{ 285768} {49} =5832$$Figure 5 shows a plot of these numbers.

Figure 5: permalink

These numbers define OEIS A240520 that clearly goes on forever:

A240520

Numbers that set a new integer record for the ratio between the product and the sum of their digits

Interestingly, if the search is extended for numbers up to two million, only one new number is added and that is 1899999 with a digit product of 472392 and a sum of 54, giving the integer 8748 as the result.

Extending the search to three million, we get four new numbers.

Figure 6: permalink

The plot of the full range of numbers is shown in Figure 7.

Figure 7: permalink

What we see in Figure 7 is that 1899999 is very much a singleton, sitting there is splendid isolation. This is even more evident when the search is extended to four million. Three new numbers appear (Figure 8) but they are close together (Figure 9). 

Figure 8: permalink

Figure 9: permalink