Revisiting Hyperperfect Numbers

I made a post titled Multiperfect, Hyperfect and Superperfect Numbers on the 24th July 2019 and another titled Hemiperfect Numbers on the 3rd January 2021. These terms can be confusing and these two posts require a careful reading in order to clarify the distinctions between the types of numbers. In today's post, I'm just revisiting the hyperperfect numbers.

The reason for my revisit is that my diurnal age today, 26977, qualifies it for membership in OEIS A034897:

A034897

Hyperperfect numbers: \(n\) such that \(n = 1 + k(\sigma(n)-n-1)\) for some \(k > 0\).

To quote from the first mentioned blog post:

A number \(n\) is called \(k\)-hyperperfect if$$n=1+k \, \sum_i d_i=1+k \,(\sigma(n)-n-1)$$where \( \sigma(n)\) is the divisor function and the summation is over the proper divisors with \(1<d_i<n\). Figure 1 shows a table of the first few hyperperfect numbers where \(k=1\) returns the perfect numbers:

Figure 1

What OEIS A034897 shows however, are all the hyperfect numbers regardless of the value of \(k\). Figure 2 shows a table of the initial hyperperfect numbers together with their corresponding \(k\) values.

Figure 2: permalink

Here is a fuller list of hyperperfect numbers but without \(k\) values, up to those that are a little over one million:

6, 21, 28, 301, 325, 496, 697, 1333, 1909, 2041, 2133, 3901, 8128, 10693, 16513, 19521, 24601, 26977, 51301, 96361, 130153, 159841, 163201, 176661, 214273, 250321, 275833, 296341, 306181, 389593, 486877, 495529, 542413, 808861, 1005421, 1005649, 1055833

As can be seen, 226977 is a 48-hyperperfect number and the last hyperperfect number that I will encounter in my lifetime (when considering my diurnal age). The next one is 51301 by which time I will be long gone. Thus for 26977 we have:$$ \begin{array} 226977 &=1+48 \times \, \sum_i d_i\\ &=1+48 \times \,(\sigma(26977)-26977-1)\\ &=1+48 \times (27540-26977-1)\\ &=1+48 \times 562\\&=26977 \end{array} $$Here is a link to the Wikipedia article about hyperfect numbers. 

Multiperfect, Hyperfect and Superperfect Numbers

Today I stumbled upon the term abundancy, as applied in a mathematical sense to the abundancy of a number and defined as the ratio:$$ \frac{\sigma(n)}{n} \text{ where } \sigma(n) \text{ is the divisor function}$$For \(n=1, 2, ...\), the first few values are:$$1, 3/2, 4/3, 7/4, 6/5, 2, 8/7, 15/8, 13/9, 9/5, 12/11, 7/3, 14/13, ...$$It's interesting to look at the approximate decimal value of this ratio as the numbers range from one up to a million. My analysis using SageMathCell revealed the following records for the maximum abundancy:

• 60         -->     14/5 \( \approx \) 2.80000000000000 between 1 and 100
• 840        -->   24/7 \( \approx \) 3.42857142857143 between 1 and 1000
• 5040      -->   403/105 \( \approx \) 3.83809523809524 between 1 and 10000
• 55440    -->   1612/385 \( \approx \) 4.18701298701299 between 1 and 100000
• 720720  -->   248/55 \( \approx \) 4.50909090909091 between 1 and 1000000

If this ratio turns out to be a positive integer, then \(n\) is said to be a multiperfect number. The first few are:$$1, 6, 28, 120, 496, 672, 8128, ...$$corresponding to the abundancies of: $$1, 2, 2, 3, 2, 3, 2, 4, 4, ... $$So a formal definition of a multiperfect number might be that a number \(n\) is \(k\)-multiperfect (also called a \(k\)-multiply perfect number or \(k\)-pluperfect number) if: $$ \sigma(n)=kn \text{ for some integer } k \geq 2$$The value of \(k\) is called the class. The special case \(k=2\) corresponds to perfect numbers \(P_2\). Source. Figure 1 shows the first few examples of such classes with the second column representing the associated OEIS reference:

Figure 1

Let's move on to hyperperfect numbers. A number \(n\) is called \(k\)-hyperperfect if$$n=1+k \, \sum_i d_i=1+k \,(\sigma(n)-n-1)$$where \( \sigma(n)\) is the divisor function and the summation is over the proper divisors with \(1<d_i<n\). Source. Figure 2 shows a table of the first few hyperperfect numbers:

Figure 2

As can be seen, the \(k\)-hyperperfect numbers reduce to the perfect numbers when \(k=1\) and the multiperfect numbers when \(k=2\). For other values of \(k\), the two sets of numbers differ. 

Just to confuse matters further, Figure 3 shows a diagram with other types of numbers that are related to abundance. As can be seen, the terms perfect, colossally abundant, superior highly composite, superabundant, highly composite, primitive abundant, highly abundant, deficient and abundant are used. 

Figure 3: source

Some of these I'm already familiar with and relevant posts include:

• primitive abundant numbers on 15th December 2018
• abundant numbers on 18th January 2017
• odd primitive abundant numbers on 21st May 2107
• zumkellar, half zumkellar and pseudoperfect numbers on 22nd November 2018
• admirable numbers and compatible numbers on 4th December 2018
• unitary divisors on 16th May 2018
• deficient numbers on 28th January 2018

In following the source of the image that I used in Figure 3, I came across another class of numbers called superperfect numbers. A superperfect number is defined by this source as a positive integer \(n\) that satisfies:$$\sigma^2(n)=\sigma(\sigma(n))=2n$$The first few superperfect numbers are:$$2, 4, 16, 64, 4096, 65536, 262144, 1073741824, ...$$which is OEIS A019279. The comment is made that:

If \(n\) is an even superperfect number, then \(n\) must be a power of \(2\), \(2k\), such that \(2^{k+1} − 1\) is a Mersenne prime. It is not known whether there are any odd superperfect numbers. An odd superperfect number \(n\) would have to be a square number such that either \(n\) or \( \sigma(n) \) is divisible by at least three distinct primes. There are no odd superperfect numbers below \(7×10^{24}\). 

Perfect and superperfect numbers are examples of the wider class of \(m\)-superperfect numbers, which satisfy:$$ \sigma^m(n)=2n \text{ with }m=1 \text{ and } 2$$For \(m \geq 3\) there are no even \(m\)-superperfect numbers. The \(m\)-superperfect numbers are in turn examples of \((m,k)\)-superperfect numbers which satisfy:$$ \sigma^m(n)=kn$$With this notation, perfect numbers are \((1,2)\)-perfect, multiperfect numbers are \((1,k)\)-perfect, superperfect numbers are \((2,2)\)-perfect and \(m\)-superperfect numbers are \((m,2)\)-perfect numbers. Examples are shown in Figure 4.

Figure 4

I must confess that I head is spinning with all this nomenclature so I'll leave off there.