23500: Blast from the Past

One of my earliest posts regarding my diurnal age dates to August 5th 2013 and was uploaded to my Pedagogical Posturing blog. This Mathematical Meandering blog did not exist at that time. This was before I had any notion that OEIS and other resources existed. Rather naively, I wrote in a post titled Reflections on 23500:

Today I'm 23500 days old and I was looking around to see if there was any significance to this number. It's factors are unremarkable (2^2 x 5^3 x 47) but the fact that it's halfway between 23000 and 24000 means that it pops up quite frequently in Internet searches, as would 22500 or 24500 I would imagine. A search reveals that the approximate population of Boston in 1620 was 23500 and there are several towns around the world that are listed as having this population currently e.g. Bishopbriggs in Scotland.

Bishopbriggs grew from a small rural village on the old road from Glasgow to Kirkintilloch and Stirling during the 19th century, eventually growing to incorporate the adjacent villages of Auchinairn, Cadder, Jellyhill and Mavis Valley. It currently has a population of approximately 23,500 people.

Source: http://en.wikipedia.org/wiki/Bishopbriggs

It turns out that Mount Isa has the same population (source):

Mount Isa is located just 200 kilometres from the Northern Territory border and 1,829 kilometres from Queensland’s capital, Brisbane. The nearest major city, Townsville, can be found 883 kilometres from The Isa. Mount Isa covers an area of over 43,310 square kilometres, making it geographically the second largest city in Australia to Kalgoorlie-Boulder, Western Australia ... With a population of approximately 23,500, Mount Isa is a major service centre for north-west Queensland.

Many other examples of towns having populations of about 23500 could be quoted. In addition of populations, the number sometimes comes up as a dollar figure (source):

In its third annual funding cycle, the Black Philanthropy Initiative has pumped $23,500 back into the Winston-Salem area to help African Americans improve their parenting skills.

Interestingly, it turns out that the centre of the Sun is about 23500 times more distant from us than the centre of the Earth (source).

The sun is far enough away (about 23,500 earth radii) that it took a long time before people knew accurately how far away the sun was. Certainly the ancient Greeks had calculated the distance, but they also knew that their results could be off. 

Many countries in the world have five digit postal codes or zip codes as they are sometimes known. These codes identify particular locations within the country e.g. Muang Prachinburi, Prachinburi, Thailand has a postcode of 23500. The United States uses a five digit system but apparently there is no location corresponding to 23500, although there is for 23499 and 23501. 

I've improved the formatting somewhat but that post was made almost eight years ago and is testimony to how little I knew about number properties at the time. With my current skill set, what more can I uncover about 23500? It turns out that it has 18 entries in the OEIS and a couple of them are interesting enough for comment.

The first is its membership of OEIS A133524:

A133524

Sum of squares of four consecutive primes.

The members of the sequence, up to 23500, are:

87, 204, 364, 628, 940, 1348, 2020, 2692, 3700, 4852, 5860, 7108, 8548, 10348, 12220, 14500, 16732, 18580, 21100, 23500

Nowadays I'd write a program in SageMath to generate this sequence. Here is a permalink to that program in SageMathCell.

The second is its membership of OEIS A035959 where \(n=42\):

A035959

Number of partitions of \(n\) in which no parts are multiples of 5.

The members of the sequence, up to 23500, are:

1, 1, 2, 3, 5, 6, 10, 13, 19, 25, 34, 44, 60, 76, 100, 127, 164, 205, 262, 325, 409, 505, 628, 769, 950, 1156, 1414, 1713, 2081, 2505, 3026, 3625, 4352, 5192, 6200, 7364, 8756, 10357, 12258, 14450, 17034, 20006, 23500

Here is the permalink to the algorithm in SageMathCell that will generate this sequence. 

As we turn to Numbers Aplenty, my other major source of information about numbers, we discover that it is a Harshad number as well as an abundant, practical, pseudoperfect and Zumkeller number. There's lots more information there as well. See Figure 1.

Figure 1: link

Over the years, I've developed programs related to practical, pseudoperfect and Zumkeller numbers. I've stored these as a Google Document accessible via this link.

For example, here is a permalink to a program that will determine all the practical numbers in a given range. Remember a practical number has the property that each smaller number is the sum of distinct divisors of that number. Between 23450 and 23550, there are the following practical numbers:

23450, 23452, 23460, 23472, 23474, 23478, 23484, 23490, 23496, 23500, 23504, 23506, 23520, 23528, 23530, 23532, 23540, 23542, 23544, 23548, 23550

A pseudoperfect number, sometimes called a semiperfect number, is a number in which a subset of its proper divisors sum to the number itself. Nearly all abundant numbers are pseudoperfect and the ones that aren't are called weird. In the case of 23500, there are many such subsets that can be formed from its proper divisors of 1, 2, 4, 5, 10, 20, 25, 47, 50, 94, 100, 125, 188, 235, 250, 470, 500, 940, 1175, 2350, 4700, 5875 and 11750. One such subset is 1175, 4700, 5875 and 11750. Here is permalink to SageMathCell and a program that will display all of the possible subsets.

Zumkeller numbers are related to perfect numbers such as 6, the divisors of which can be written as the set {1, 2, 3, 6}. There are two mutually exclusive subsets of this set, {1, 2, 3}  and {6}, whose union is the original set and both of whose elements add to 6. Zumkeller numbers are similar in that there are two mutually exclusive subsets of this set of divisors, whose union is the original set but with the difference that both of the elements in each subset add to a number other than the originating number. For example, 20 has the set of divisors {1, 2, 4, 5, 10, 20} and there are two subsets {1, 20} and {2, 4, 5, 10} that both total 21. Thus 20 is a Zumkeller number. In the case of 23500, there are many such pairs of subsets. Here is one example:

[4, 10, 94, 250, 2350, 23500] and [1, 2, 5, 20, 25, 47, 50, 100, 125, 188, 235, 470, 500, 940, 1175, 4700, 5875, 11750] which both sum to 26208.

Here is a permalink to a SageMathCell program that will display all the pairs of subsets.

Harshad or Niven numbers are divisible by the sum of their digits. Here is a permalink to a SageMathCell program that displays all such numbers in the range from 23450 to 23550. These numbers are:

23450, 23454, 23457, 23460, 23472, 23490, 23496, 23500, 23505, 23508, 23520, 23526, 23530, 23544, 23550

23500 is formed from a concatenation of the first three prime numbers (2, 3 and 5), followed by two zeroes. It's interesting to note that 23500 written in scientific notation as \(2.35 \times 10^4\) features all the digits from 0 to 5. Earth's axial tilt is close to 23.5° and, as mentioned in my original post, there is this most interesting astronomical detail regarding 23500: 

Distance to Sun = 1.5 x 10^13 cm = 23500 Earth radii = 1 Astronomical Unit (AU) 

So nothing mathematically Earth-shattering in this post but just a reflection on the differences in my mathematical knowledge between now and then.