Hemiperfect Numbers

The so-called hemiperfect numbers relate a concept called abundancy that I've dealt with in two previous posts:

• Multiperfect, Hyperperfect and Superperfect Numbers (July 24th 2019) 
• Friendly versus Solitary Numbers (December 16th 2020)

I'll define the abundancy of a number $n$ once again to be the ratio of the sum-of-divisors of $n$ to $n$ itself. It is given by the formula:$$\frac{{\sigma }_1(n)}{n}\text{ where }{\sigma }_1(n)\text{ is the divisor function}$$Note that abundancy may also be defined as:$${\sigma }_{-1}(n)\text{ where }{\sigma }_{-1}(n)\text{ represents the sum of the reciprocals of the divisors of }n$$A multiperfect (sometimes multiply perfect) number is a number whose abundancy is a whole number:$$\frac{{\sigma }_1(n)}{n}=k\text{ with }k\text{ an integer }\ge 2$$We can refer to such a number as $k$-perfect with 2-perfect numbers being the perfect numbers 6, 28, 496, 8128 etc.

Today I turned 26208 days old and discovered that this number is a member of OEIS A159907:

A159907

Numbers $n$ with half-integral abundancy index such that:                  $$\frac{{\sigma }_1(n)}{n}=k+\frac{1}{2}\text{ with integer }k$$

Numbers of this sort are termed hemiperfect. The sequence, up to 26208, consists of 2, 24, 4320, 4680, 26208 where:

$$\begin{array}{rl}\frac{{\sigma }_1(2)}{2}& =\frac{3}{2}=1+\frac{1}{2}\\ \frac{{\sigma }_1(24)}{24}& =\frac{60}{24}=2+\frac{1}{2}\\ \frac{{\sigma }_1(4320)}{4320}& =\frac{15120}{4320}=3+\frac{1}{2}\\ \frac{{\sigma }_1(4680)}{4680}& =\frac{16380}{4680}=3+\frac{1}{2}\\ \frac{{\sigma }_1(26208)}{26208}& =\frac{91728}{26208}=3+\frac{1}{2}\end{array}$$After this the numbers get bigger quickly. The next hemiperfect number is 8910720 which has an abundancy of 9/2 and is termed 9-hemiperfect. Similarly, 2 is 3-hemiperfect, 24 is 5-hemiperfect and 4320, 4680 and 26208 are 7-hemiperfect.