Odd Deficient Numbers

I've written about deficient numbers in an eponymous post on January 28th 2018 and in which I mentioned, for the first time in my postings, the ratio between the sum of the divisors of a number and the number itself viz. \( \displaystyle \frac{\sigma_1(n)}{n} \).

In that post, I didn't refer to the ratio by its name of abundancy but in later posts I explored the concept of abundancy in more detail. Here are links to posts in which it was discussed:

• Multiperfect, Hyperperfect and Superperfect Numbers on July 24th 2019
• Friendly versus Solitary Numbers on December 16th 2020
• Hemiperfect Numbers on January 3rd 2021

The last two posts, as can be noted, are relatively recent. Today, in turning 26325 days old, I encountered a reference to abundancy once again. Specifically in the context of OEIS A188597:

A188597

Odd deficient numbers whose abundancy is closer to 2 than any smaller odd deficient number.

My diurnal age is a member of this sequence which runs:

1, 3, 9, 15, 45, 105, 315, 1155, 26325, 33705, 449295, 1805475, 10240425, 13800465, 16029405, 16286445, 21003885, 32062485, 132701205, 594397485, 815634435, 29169504045, 40833636525, 295612416135, 636988686495, 660733931655, 724387847085, 740099543085, 1707894294975, 4439852974095, 7454198513685

 Figure 1 shows the progression:

Figure 1

Not surprisingly, all these numbers are highly composite, despite all being deficient. This can be seen in Figure 2 where a table of factorisation and number of divisors is presented.

Figure 2

There are a number of related sequences, one of them is OEIS A171929. 

A171929

Odd numbers whose abundancy is closer to 2 than any smaller odd number.

Here the numbers need only to odd and can be abundant or deficient. Figure 3 shows the abundancy and its absolute difference from 2.

Figure 3

Another related sequence is OEIS A188263:

A188263

Odd abundant numbers whose abundancy is closer to 2 than any smaller odd abundant number.

 

In this sequence, all the numbers are abundant and thus their abundancy is greater than 2. Figure 4 shows a table of the initial numbers and their abundancies.

Figure 4

It's interesting to consider the idea that the limit of the abundancy of these sorts of odd abundant and odd deficient numbers is actually 2 as their abundancy can be as close to 2 as desired.