Every Number Is Interesting

It's been quite some time since I last posted. My temporary relocation to the wilds of Sumatra has hampered my blogging opportunities but I wanted to make at least one post before July is over. It's July 31st now but on July 29th I turned 26780 days old. Initially, I could find little of interest about this number but I knew this was only because I wasn't looking hard enough. 

Once I did I soon realised that 26780 is special in at least one way, namely none of its digits repeat. This led me to ask how many such five digit numbers have this property? The five digit numbers range from 10000 to 99999 and there are thus 89999 of them. If there are to be five digits and none of them can repeat then the first, or leftmost digit, cannot be zero so there are 9 possibilities. For the second digit there are also 9 possibilities, followed by 8, 7 and 6 for the remaining positions, giving at total of 9 x 9 x 8 x 7 x 6 = 27216 numbers. This represents 30.24% of the range. 

Thus 26780 is special but not that special because about three out of every ten five digit numbers have this same property. Examining the number further, I noticed that the digit sum of 26780 is 23 and thus it has a digital root of five. Thus it's a five digit number with a digital root of five. Numbers with these two properties are ten times as rare. In fact there are 3024 of them giving a percentage of 3.36%. This makes the number a lot more special.

On still further investigation, I noticed that 26780 has five prime factors, counting multiplicity, because it factorises to 2 x 2 x 5 x 13 x 103. There are only  253 such numbers out of the 89999 and this represents 0.0028%, making the number something of a rara avis. Here is the list of such numbers:

10256, 10625, 10832, 12056, 12650, 13064, 13208, 13496, 14036, 14360, 14576, 14630, 14756, 14792, 15260, 15368, 15620, 15728, 15980, 16304, 16340, 16952, 17204, 17240, 17384, 17420, 17456, 17528, 17960, 18536, 18590, 19256, 19472, 19580, 19652, 19832, 20984, 23180, 23450, 23540, 23576, 23864, 24368, 25016, 25340, 25916, 25970, 26780, 27140, 27536, 27860, 27896, 27950, 28175, 28490, 28760, 28976, 29768, 29876, 30416, 30758, 30875, 31064, 31208, 31496, 31568, 31784, 31820, 31892, 31928, 32450, 32504, 34250, 34520, 34790, 34916, 34952, 35096, 35240, 36140, 36248, 36752, 37625, 37940, 38012, 38120, 38570, 39056, 39128, 39416, 39650, 39704, 39875, 40136, 40172, 40568, 40712, 41756, 41936, 42980, 43016, 43250, 43610, 45032, 45392, 46508, 46580, 46832, 47012, 47138, 47192, 47912, 48650, 48920, 49028, 49352, 49532, 49820, 50468, 50648, 50792, 51260, 51620, 51980, 52016, 52430, 53168, 53690, 53924, 54230, 54392, 54608, 54680, 54860, 56012, 56210, 56408, 56912, 56984, 57128, 57380, 58190, 58460, 58712, 59180, 59216, 59720, 59864, 60125, 60152, 60872, 61304, 62150, 62348, 62510, 62780, 63752, 63824, 64130, 65048, 65192, 65219, 65480, 67208, 67298, 67928, 68072, 68324, 68450, 68540, 69125, 69350, 69584, 70196, 71825, 71852, 72590, 72968, 73508, 73580, 73625, 73850, 74192, 75164, 75416, 75803, 76280, 76532, 76820, 79160, 79250, 79268, 79304, 79430, 80132, 80276, 80456, 80465, 81032, 81356, 81392, 81752, 81950, 82076, 82760, 82904, 83912, 84632, 85316, 85640, 85910, 86504, 86792, 87125, 87620, 89420, 89456, 89672, 89726, 90248, 90356, 90428, 91364, 91472, 91580, 91832, 91850, 92876, 93128, 93560, 93740, 94280, 94316, 94352, 94820, 94856, 95108, 95432, 95612, 95720, 95810, 95864, 96278, 96350, 96530, 96728, 97016, 97250, 97340, 97682, 98150, 98456, 98735

Such numbers might well find a place in the Online Encyclopedia of Integer Sequence (OEIS) but I'll not be proposing them as a sequence given that I don't contribute anymore. The sequence can be stated as follows:

Five digit numbers in which no digit is repeated, the digital root is 5 and there are five prime factors (not necessarily distinct)

The reasons I've given for not interacting with the OEIS are stated in an earlier post. The point of this post is to show that every number has interesting properties if you look hard enough.

ADDENDUM: October 19th 2022

I had to include the following that I discovered here as it's relevant to the content of this post. 

Folklore tells us that there are no uninteresting natural numbers. The argument hinges on the following observation: Every subset of the natural numbers is either empty, or has a smallest element. The argument usually goes something like this. If there would be any uninteresting natural numbers, the set U of all these uninteresting natural numbers would have a smallest element, say u ∈ U. But u in itself has a very remarkable property. u is the smallest uninteresting natural number, which is very interesting indeed. So U, the set of all the uninteresting natural numbers, can not have a smallest element, therefore U must be empty. In other words, all natural numbers are interesting.