World of Numbers

Today I came across an interesting website via a link in an OEIS entry for 27454, the number associated with my diurnal age as of today's date. Figure 1 shows a screenshot.

Figure 1

The link provided to P. De Geest's  Nine Digits Digressions takes us to a particular page on the World of Numbers website. Figure 2 shows the page and Figure 3 shows the home page of the website.

Figure 2

Figure 3

Looking at the website, I immediately thought that it was one of those websites that had been created in the 1990s and then abandoned. However, a closer look showed that it had been created in 1996 but updated on June 2nd 2024 which is the date on which I'm creating this post. So remarkably the site has been maintained from 1996 to 2024 by P. De Geest.

So who is P. De Geest? Well his site provides a not-so-recent photo and a brief bio:

Photo taken in 2004

E-mail: pdg@worldofnumbers.com 

Web Page: http://www.worldofnumbers.com/index.html 

My name is Patrick De Geest, born on the 9th of October 1956, in Wezembeek-Oppem, Belgium (about 10 km east of Brussels), unmarried, mildly myopic, graduated in architecture but never practiced the profession. Currently I'm an employee working in the aircargo export sector (National Airport Zaventem). I didn't lose my interest in beautiful patterns and proportions though, and managed to transfer it to the field of numbers. 

Also, through the years, I gradually became familiar with the use of personal computers (no, I'll never sell my first Sinclair ZX81) and learned for programming techniques (basic, assembly, ...). All these 'creativities' culminated recently in a website about recreational mathematics with 'palindromes' as the main topic. I opted for palindromes not because of my length (181 cm), my average weight (77 kg) or my housenumber (141) but because I was attracted by their overall symmetry and the fact that it was a novel and thus insufficiently studied subject. Thanks to many contributors from all over the world the site is still expanding. 

For the rest I'm a rather quiet individual who likes to read an occasional book, watch a movie, listen to classical music, travel once or twice a year to a near/far exotic  destination and bike from time to time when the weather permits.

Anyway the point is that the site contains a wealth of information about curious number properties with Patrick giving the following overview of the site's contents:

In this well-filled website you'll find a multitude of facts and figures about topics from the  World!Of Numbers . Don't look for a logical order. It is an amalgamation of randomly gathered numbers, curios, puzzles, palindromes, primes, gems, your much valued contributions and more general information. Enjoy! Patrick De Geest  

Like Taneja's papers described in my previous post, there is great content here for future posts to this blog. Getting back to the original OEIS sequence, we see that:

\(27454^ {0.25} = 12.\overline{87215934}68573\)

The first nine digits of the decimal part do indeed contain all the digits from 1 to 9. Interestingly I can find no reference to these sorts of calculations of page 7 of "Nine Digits" topic. Perhaps it's on one of the other pages. Numbers like 27454 are part of OEIS A034279:

A034279

Decimal part of \(a(n)^{1/4}\) starts with a 'nine digits' anagram.

The sequence begins: 7396, 8751, 8933, 8950, 9070, 11184, 26484, 26522, 27454, 30858, 36923, 39895, 40828, 42793, 47311, 58738, 58985, 61143, 72788, 73506, 75636, 79562, 80138, 80260, 81101, 83261, 94796, 96256, 101915, 102189, 103310, 103416, 108901

There's no reason to restrict ourselves to the fourth root and there are sequences corresponding to numbers raised to  1/2, 1/3, 1/5, 1/6, 1/7 and 1/8 powers and probably more. Here is a permalink to a general purpose algorithm that will generate sequences for any power desired. The relevant OEIS sequences are:

• square root: OEIS A034277 with initial members being 86, 868, 1278, 5211, 7494, 7772, 14567, 17573, 18421, 20844, 24960, 26535, 29172, 29301, 29987, 32845
  
  
• cube root: OEIS A034278 with initial members being 429, 939, 7015, 11456, 15221, 17521, 21000, 21160, 22397, 24789, 28916, 30945, 33743, 35440, 36732
  
  
• fifth root: OEIS A034280 with initial members being 12, 1635, 2112, 6905, 15376, 18660, 18795, 20085, 21086, 21447, 22064, 23077, 23540, 25817, 27040, 28204, 30668, 31258, 31287, 37407, 38533
  
  
• sixth root: OEIS A034281 with initial members being 648, 695, 1979, 7509, 9214, 12567, 19740, 21555, 24235, 24646, 25624, 27427, 30717, 30748
  
  
• seventh root: OEIS A034282 with initial members being 551, 574, 2998, 8265, 9407, 10357, 12459, 15885, 20480, 26103, 26134, 29297, 35096, 35984, 37113, 39084, 39733, 39735
  
  
• eighth root: OEIS A034283 with initial members being 3927, 4176, 10041, 10827, 13575, 15544, 15853, 17244, 20154, 24759, 25146, 30008, 30038, 30635, 30692, 32046, 37215

That's enough I think. Remember that there are factorial 9 ways to arrange the nine digits and this equals 362880, an impressive number of permutations.