What's Special About 28046?

Figure 1

Figure 1 shows a screenshot of a post that I uploaded on the 27th September 2022 in which I noted that the number 26840 contained all the even digits exactly once. Today I turned 28046 days old and this number is the next such number after 26840 with this property. There are only 96 such numbers overall and as I also noted in my blog post at the time:

"I'll only see another six such numbers in my lifetime: 28046, 28064, 28406, 28460, 28604, 28640."

THE 28046 MONUMENT: A COLOSSAL ACHIEVEMENT

So, one down and five to go with the next one only 18 days away. After that, there's a gap of slightly less than a year. The property of these numbers is very much base dependent of course but nonetheless interesting from a recreational mathematical perspective.

A Special Number: 3367

 Figure 1 shows an interesting result via Cliff Pickover.

To see why 3367 can be written as the various fractions shown, one approach is to consider a number of the form \( 0 \, a \, 0 \, a \, 0 \, a\) where \(1 \geq a \geq 99\). Let's manipulate this number in the following way:$$  \begin{align} \frac{0 \, a \, 0 \, a \, 0 \, a}{a+a+a} &= \frac{a \times 0 \, 1 \, 0 \, 1 \, 0 \, 1}{3 \, a}\\ \\ &= \frac{0 \, 1 \, 0 \, 1 \, 0 \, 1}{3} \\  \\ &=3 \, 3 \, 6 \,7 \end{align}$$Thus when \(a=11\) we have:$$\frac{0 \, a \, 0 \, a \, 0 \, a}{a+a+a} = \frac{1 \, 1 \, 1 \, 1 \, 1 \, 1}{11 + 11 +11}$$If \(a>99\), then the pattern breaks down.

Unremarkable Numbers

Every now and again, as I track my diurnal age, I come across a number that is, for want of a better word, unremarkable. 26534 appears to be such a number. It has two entries in the OEIS and both of them are of the inscrutable variety. Searches elsewhere reveal nothing meaningful either. 

Such apparent "duds" can serve as a stimulus to really look at the number more closely in an effort to extract some properties of interest. In the case of 26534, two properties came to light when I did this. These are:

• each of its five digits is unique (2, 6, 5, 3, 4)
• the digits when ordered are sequential (2, 3, 4, 5, 6)

This becomes important when tracking my diurnal age because, in the span of 10000 numbers from 20000 to 29999, there are very few numbers with this property. There are 120 permutations of the five digits (2, 3, 4, 5, 6) but if we fix the 2 in place then there are only 24 numbers that lie in this range from 20000 to 29999. These are:

23456, 23465, 23546, 23564, 23645, 23654, 24356, 24365, 24536, 24563, 24635, 24653, 25346, 25364, 25436, 25463, 25634, 25643, 26345, 26354, 26435, 26453, 26534, 26543

As can be seen, 26534 is the penultimate number when such numbers are ordered and the final number, 26543, is only nine days away. Another group of 24 numbers can be found for the range from 20000 to 29999 using the digits 1, 2, 3, 4, 5 by fixing the 2 in first place. The numbers in this group are:

21345, 21354, 21435, 21453, 21534, 21543, 23145, 23154, 23415, 23451, 23514, 23541, 24135, 24153, 24315, 24351, 24513, 24531, 25134, 25143, 25314, 25341, 25413, 25431

Using the digits from 0 to 4 and again fixing the 2 in first place, we get another 24 digits in the range from 20000 to 29999. These are:

20134, 20143, 20314, 20341, 20413, 20431, 21034, 21043, 21304, 21340, 21403, 21430, 23014, 23041, 23104, 23140, 23401, 23410, 24013, 24031, 24103, 24130, 24301, 24310

Altogether then there are 72 numbers in the range from 20000 to 29999 that meet the criteria:

• each digit in unique
• the digits when ordered are sequential

Thus, out of 10000 numbers, only 0.72% satisfy the criteria. It turns out that 26534 is rather special after all. Here is a permalink that can be used to generate these numbers.