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.