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.