Arithmetic Numbers

I was familiar with the concept of an arithmetic derivative but today I was reminded of the concept of an arithmetic number. I encounter the term almost daily when referring to the website Numbers Aplenty but hadn't paid it much attention. Here is a definition from Wikipedia:

In number theory, an arithmetic number is an integer for which the average of its positive divisors is also an integer. For instance, 6 is an arithmetic number because the average of its divisors is 3:$$ \frac{1+2+3+6}{4}=3$$Phrasing it more rigorously, we could say that a number \(n\) is arithmetic if the number of divisors \(d(n)\) or \( \sigma_0(n) \) divides the sum of divisors \( \sigma_1(n) \).

This ratio is not be confused with the ratio that I was investigating in my previous post on Barely Abundant Numbers. In that case, the ratio was \( \sigma(n) \) over \(n\) which determines whether a number is deficient, perfect or abundant. If the ratio is less than 2, the number is deficient. If it's greater than 2, the number is abundant. We know 6 is a perfect number because the ratio equals 2:$$ \frac{1+2+3+6}{6}=2$$The term was brought to my attention by one of the properties of the number associated with my diurnal age of 26744. The property qualities the number for membership in OEIS A107924 as the 119th member.

A107924

Even numbers \(n\) such that \(n^2\) is an arithmetic number.

296, 536, 632, 872, 1208, 1304, 1544, 2072, 2216, 2648, 2984, 3584, 3656, 3752, 3848, 3896, 3904, 3992, 4328, 4424, 4568, 4904, 5624, 5672, 5912, 6008, 6104, 6584, 6968, 7016, 7256, 7352, 7928, 8216, 8264, 8456, 8696, 8896, 8936, 9032, 9128, 9176, 9368, 9608, 9704, 10184, 10376, 10616, 10808, 11048, 11336, 11384, 11624, 12008, 12392, 12632, 12728, 12968, 13304, 13504, 13976, 14072, 14312, 14408, 14472, 14648, 14984, 15512, 15704, 15992, 16088, 16424, 16568, 16616, 16664, 16952, 17064, 17096, 17432, 17768, 18008, 18056, 18344, 18536, 18776, 19016, 19112, 19592, 19784, 19832, 20024, 20072, 20456, 20888, 21368, 21464, 21608, 21704, 22136, 22376, 22808, 22952, 23048, 23384, 23488, 23872, 24152, 24392, 24488, 24776, 25496, 25592, 25736, 25832, 26072, 26168, 26408, 26504, 26744

The OEIS comments state that "odd numbers with this property are much more numerous" and this is indeed true. In the same range (up to 26744) there are 2462 such odd numbers. Checking for \(n=26744\), we find that \( n^2=715241536 \) and we have: $$ \frac{\sigma(715241536,1)}{\sigma(715241536, 0)}=\frac{1419732111}{21}=67606291$$It can be noted that 26744 is itself an arithmetic number but this is hardly surprising as most numbers are and this includes all odd primes. To see this, take any odd prime \(p\) with the sum of its two divisors being \(1+p\), an even number and thus divisible by 2. According to Numbers Aplenty, \(p^p\) is also an arithmetic number.

It is possible to find quite long runs of consecutive arithmetic numbers. For example, one of length 105 starts at 3033935561. The smallest 3 × 3 magic square made of consecutive arithmetic numbers is shown below:

The 10000th arithmetic number is 12953. This gives a good idea of their frequency and why numbers that aren't arithmetic numbers are more interesting because they are less frequent. For example, \(84=2^2 \times 3 \times 7\) has twelve divisors 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84 that sum to 224. It is not an arithmetic number because 12 does not divide 224 without remainder.

Just to confuse matters, \(26744 = 2^3 \times 3343\) is a tau number, defined as a number that is divisible by the number of its divisors (in this case, eight). If we put all this together and ask what even numbers \(n\) have the properties that:$$ \begin{align} \frac{ \sigma(n^2,1)}{\sigma(n^2,0)}&=k_1\\ \frac{ \sigma(n,1)}{\sigma(n,0)}&=k_2\\ \frac{ n}{\sigma(n,0)}&=k_3 \end{align}$$where \(k_1, k_2, k_3\) are integers. It turns that the list is fairly short in the range up to 26744: 

632, 1208, 3896, 6008, 6584, 7352, 7928, 8696, 10616, 11384, 13304, 14072, 14648, 15992, 20024, 21368, 22136, 26168, 26744