## My Vision for Number Gossip

I run Number Gossip, where you can input a number and get some of its cute properties. For example, the number 63 is composite, deficient, evil, lucky, and odd. In addition, it has a unique property: 63 is the smallest number out of two (the other being 69), such that the common alphabetical value of its Roman representation is equal to itself. Indeed, the Roman representation of 63 is LXIII, where L is the 12th digit, X is the 24th, and I is the 9th. Summing them up, we get 12 + 24 + 9 + 9 + 9 = 63 — the number itself.

I have a list of about 50 properties of numbers that my program checks. Each number greater than one gets at least four properties. This is because I have four groups of properties that cover all the numbers. Every number is either odd or even. Every number is either deficient, perfect, or abundant. Every number greater than one is either prime or composite. Every number is either evil or odious.

In addition, I collect unique number properties. During the website’s conception, I decided not to list all possible unique properties that I could imagine but to limit the list to interesting and unusual properties. My least favorite properties of numbers are the ones that contain many parameters. For example, 138 is the smallest number whose base 4 representation (2022) contains 1 zero and 3 twos. If you are submitting a number property to me, keep this in mind.

Some parameters are more forgiving than others. For example, 361 is the smallest number which is not a multiple of 9, whose digital sum coincides with the digital sum of its largest proper divisor. In more detail, the digital sum of 361 is 3 + 6 + 1 = 10, while 19, the largest divisor of 361, has the same digital sum of 10. In this case, the parameter 9 is special: for a multiple of 9, it is too easy to find examples that work, such as 18, 27, and so on. Sequence A345309 lists numbers whose digital sum coincides with the digital sum of their largest proper divisor. The first 15 terms of the sequence are divisible by 9, and 361 is the smallest term that is not divisible by 9.

By the way, another number that is buried deep in a sequence is 945, which is the smallest odd abundant number. There are 231 abundant numbers smaller than 945; all of them are even.

A more recent addition to my collection is related to the sequence of distended numbers (A051772). Distended numbers are positive integers n for which each divisor of n is greater than the sum of all smaller divisors. It is easy to see that for distended numbers, all sums of subsets of divisors are distinct. The opposite is not true: 175 is the smallest number, where all sums of subsets of its divisors are distinct, but the number itself is not distended.

It is difficult to find special properties for larger numbers, so I am less picky with them. For example, 3841 is the number of intersections of diagonals inside a regular icosagon. The word icosagon hides a parameter, but I still like the property.

I invite people to submit number properties to me. And I received many interesting submissions. For example, the following oxymoronic property was submitted by Alexey Radul: 1 is the only square-free square.

The numbers below 200 that still lack unique properties are 116, 147, 155, 162, 166, 182, and 194. The earliest century that doesn’t have unique number properties ranges between 7000 and 7100. The next ones are 8000–8100 and 9100–9200. By the way, my site goes up to ten thousand.

I also have a lot of properties in my internal database that I haven’t checked yet. I am most interested in the proof of the following property: 26 is the only number to be sandwiched between any two non-trivial powers.

Share:

1. #### Jon Awbrey:

Here’s a resource for exploring many novel properties of numbers —

https://oeis.org/wiki/Riffs_and_Rotes

2. #### Jon Awbrey:

Here’s a bit of info about an interesting number I encountered recently —

https://oeis.org/wiki/Talk:Riffs_and_Rotes#59281

I noticed it had a property I dubbed “Squarefree All The Way Down” (SATWD).

3. #### Shreevatsa:

About the last line (26 is the only integer between two non-trivial powers), note that it is the n=2 case of a conjecture that for n=1 is called Catalan’s conjecture (that 8 and 9 are the only two nontrivial powers with a difference of 1) and was proved in 2002: https://en.wikipedia.org/wiki/Catalan%27s_conjecture — I don’t know whether the n=2 case has been proved or is still open, but it may be hard.

4. #### dont know if you like them or not:

ref:OEIS A350655
147
“the least positive number that can be written as p^2 + p*q + q^2 in exactly 2 ways where p and q are primes and p <= q. "

ref:OEIS A295795
155
"Largest number with exactly 22 representations as a sum of 7 positive squares."

ref:OEIS A274027
162
"Smallest number larger than 1 such that n^4 is the average of a positive cube and a positive fifth power."

5. #### tanyakh:

Thanks “don’t know”. I will add the first and the last ones to the database. The middle one is too special.