Of Substrings and Divisors

Today I turned 27375 days old and this number has an interesting property in that:$$ \begin{align} 27375 &=375 \times 73  \\ &=5 \times 73 \times 75 \end{align}$$Looking at the numbers on the RHS of the equations, it can be seen that 5, 73, 75 and 375 are all substrings of the string 27375, considering the numbers as collections of characters rather than digits. The numbers with this property form OEIS A059470:

A059470

Numbers that are the products of distinct substrings (>1) of themselves and do not end in 0.

These numbers are not numerous and up to 40000 they are:

125, 375, 735, 1197, 1296, 1352, 1593, 1734, 2346, 3125, 4224, 4872, 5775, 8448, 9072, 11715, 12768, 13455, 14476, 14673, 15625, 16128, 17136, 17493, 18432, 21168, 22176, 23184, 23391, 27216, 27375, 27648, 27864, 32256, 34272, 34398, 36288, 36864, 37296, 39375

The breakdown into divisors/substrings is as follows with some numbers having more than one representation (permalink):

125 equals the product of [25, 5]

375 equals the product of [75, 5]

735 equals the product of [35, 3, 7]

1197 equals the product of [9, 19, 7]

1296 equals the product of [9, 2, 12, 6]

1352 equals the product of [2, 52, 13]

1593 equals the product of [9, 3, 59]

1734 equals the product of [17, 34, 3]

2346 equals the product of [3, 34, 23]

3125 equals the product of [25, 125]

4224 equals the product of [24, 2, 4, 22]

4872 equals the product of [87, 7, 8]

4872 equals the product of [2, 4, 87, 7]

5775 equals the product of [75, 77]

8448 equals the product of [48, 4, 44]

9072 equals the product of [72, 9, 2, 7]

11715 equals the product of [11, 15, 71]

12768 equals the product of [2, 7, 12, 76]

13455 equals the product of [13, 3, 345]

14476 equals the product of [7, 44, 47]

14673 equals the product of [3, 73, 67]

15625 equals the product of [625, 25]

16128 equals the product of [6, 8, 12, 28]

17136 equals the product of [3, 6, 7, 136]

17493 equals the product of [17, 3, 49, 7]

18432 equals the product of [32, 4, 18, 8]

21168 equals the product of [21, 6, 168]

22176 equals the product of [176, 21, 6]

23184 equals the product of [3, 4, 84, 23]

23391 equals the product of [339, 3, 23]

27216 equals the product of [6, 21, 216]

27216 equals the product of [2, 7, 72, 27]

27375 equals the product of [375, 73]

27375 equals the product of [5, 73, 75]

27648 equals the product of [64, 2, 8, 27]

27864 equals the product of [2, 6, 86, 27]

32256 equals the product of [32, 3, 6, 56]

34272 equals the product of [3, 42, 272]

34272 equals the product of [2, 34, 7, 72]

34272 equals the product of [2, 34, 3, 4, 42]

34398 equals the product of [9, 98, 39]

36288 equals the product of [2, 3, 36, 6, 28]

36864 equals the product of [64, 3, 4, 6, 8]

37296 equals the product of [2, 37, 7, 72]

37296 equals the product of [2, 7, 296, 9]

37296 equals the product of [3, 6, 7, 296]

39375 equals the product of [3, 5, 7, 375]

I adapted the code for generating the substrings from this source (see Figure 1).

While these numbers are not frequent, there are two coming up in the relatively near future (27648 and 27864) before there is a big gap to the next number, 32256.