Dancing Digits

Whenever I'm confronted with a number associated with my diurnal age that seems to have no interesting properties, I inevitably find something very special and interesting about that number. Yesterday's number, 27542, was a number of this sort and it took me a day to stumble upon what's interesting about it.

My starting point was that it's a sphenic number because:$$2542=2 \times 47 \times 293$$Such numbers can be viewed as sphenic bricks with the three prime factors corresponding to the length, width and height. The surface area of such a brick means that there is always a second number that is inextricably linked to the original sphenic number and I've written about this in earlier posts. In the case of 27542, this second number and the surface area of the brick is 28902. This second number however, is also sphenic since we have:$$28902=2 \times 3 \times 4817$$This means that we can find the surface area of this second brick. It is 48182 which is not sphenic. However, we now have a triplet of numbers formed:$$27542, 28902, 48182$$If we find the product of these three numbers, it turns out to be an interesting number:$$27542 \times 28902 \times 48182 = 38353781868888$$It's interesting because it's 14 digits long and the digit 8 comprises precisely half of them.

The question then is how common is it for such triplets of numbers, when multiplied, to generate a number in which a single digit comprises at least 50% of all the digits? Let's reflect on the criteria for such numbers:

• the number must be sphenic and constitutes the first sphenic brick: p
• the surface area of this brick must also be a sphenic number: q
• this second number constitutes the second sphenic brick
• the surface area of this second brick constitutes the third number: r
• the product of p, q and r must contain a digit that comprises at least 50% of the digits of the number.

In the case of the digit 8, there are only three other numbers that qualify in the range up to 100,000 and these can be viewed in Figure 1. The first number in the list is 27542.

Figure 1: plethora of the digit 8

So it turns out that 27542 is the first member of a rather special sequence indeed. What about other digits? Let's start with 0.  Figure 2 shows the results for the digit 0, again up to 100,000.

Figure 2: plethora of the digit 0

The results for the digit 1 are shown in Figure 3.

Figure 3: plethora of the digit 1

For digits 2 and 3 there are no numbers and the results for digit 4 are shown in Figure 4.

Figure 4: plethora of the digit 4

For digit 6, 7 and 9 only one result is found in each case. See Figures 5, 6, 7 and 8.

Figure 5: plethora of the digit 5

Figure 6: plethora of the digit 6

Figure 7: plethora of the digit 7

Figure 8: plethora of the digit 9

Here is a permalink to the algorithm used to generate these numbers. Overall then, the numbers which produce a single digit that accounts for at least 50% of the final product of digits are:

1833, 1887, 7189, 14833, 15589, 16242, 16405, 27542, 36449, 38006, 38319, 43589, 87731

A very exclusive club indeed. Of course these number properties are base-dependent and so  fall into the realm of recreational mathematics but numberphiles are indifferent to such divisions and simply delight in the dance of the digits.