Harmonic Numbers

On January 1st 2024, I made a post titled Unitary Harmonic Numbers which are defined as numbers whose unitary divisors have a harmonic mean that is an integer. Oddly, I have never made a post simply about harmonic numbers defined as numbers whose divisors have a harmonic mean that is an integer. Like unitary harmonic numbers, they are quite rare. The number associated with my diurnal age today (27846) is one such harmonic number. These numbers make up OEIS A001599 and the initial members up to one million are (perfect numbers are shown in red):

1, 6, 28, 140, 270, 496, 672, 1638, 2970, 6200, 8128, 8190, 18600, 18620, 27846, 30240, 32760, 55860, 105664, 117800, 167400, 173600, 237510, 242060, 332640, 360360, 539400, 695520, 726180, 753480, 950976

Here is what Numbers Aplenty had to say about them:

A number \(n\)  is called harmonic divisor number if the harmonic mean of its divisors is an integer. This is equivalent to saying that the average of the divisors of \(n\) divides \(n\): $$ \frac{n}{\sigma(n)/ \tau(n)} = \frac{n \times \tau(n)}{\sigma(n)} \text{ is an integer}$$Harmonic divisor numbers are also called harmonic numbers, for brevity, or Ore numbers, after O.Ore who studied them. He proved that all the perfect numbers are also harmonic and conjectured that 1 is the only odd harmonic number. This conjecture has been verified by G.L.Cohen et al. for \(n<10^{24}\) and if true, it will imply that no odd perfect numbers exist. Jaycob Coleman has observed that all the Ore numbers up to \(10^{14}\) are also practical numbers and conjectured this holds in general. T. Goto and K. Okeya have computed a list of the 937 harmonic numbers up to \(10^{14}\).

In the case of the number (27846) associated with my diurnal age, we have:$$ \begin{align} \tau(27846) &= 48 \\ \sigma(27846) &= 78624\\ \frac{27846 \times 48}{78624} &= 17 \end{align}$$

Unitary Harmonic Numbers

As I'm creating this post it is the first day of 2024 but on the last day of 2023, I came across the term Unitary Harmonic Number for the first time. This is not surprising as they are quite rare. The initial numbers, up to 40000, are 1, 6, 45, 60, 90, 420, 630, 1512, 3780, 5460, 7560, 8190, 9100, 15925, 16632, 27300 and 31500. Yesterday, my diurnal age was 27300 which is why the term came to my attention.

A unitary harmonic number is defined as a number whose unitary divisors have a harmonic mean that is an integer. This is clearly not often the case. Let's take the number 12. It has divisors of 1, 2, 3, 4, 6 and 12. Of these, only 1, 3, 4 and 12 are unitary divisors. Let's recall that a unitary divisor of a number is a divisor such that, when divided into the number, the result is a number that has no factors in common with the divisor. For example, 2 divides into 12 to give 6 but 6 and 2 have 2 as a common factor and so 2 is not a unitary divisor. 3 however divides into 12 to give 4. 3 and 4 have no common factor and so 3 is a unitary divisor. 

Let's look at 27300. It has the following divisors:

1, 2, 3, 4, 5, 6, 7, 10, 12, 13, 14, 15, 20, 21, 25, 26, 28, 30, 35, 39, 42, 50, 52, 60, 65, 70, 75, 78, 84, 91, 100, 105, 130, 140, 150, 156, 175, 182, 195, 210, 260, 273, 300, 325, 350, 364, 390, 420, 455, 525, 546, 650, 700, 780, 910, 975, 1050, 1092, 1300, 1365, 1820, 1950, 2100, 2275, 2730, 3900, 4550, 5460, 6825, 9100, 13650, 27300

There are 32 unitary divisors of 27300 and they are:

1, 3, 4, 7, 12, 13, 21, 25, 28, 39, 52, 75, 84, 91, 100, 156, 175, 273, 300, 325, 364, 525, 700, 975, 1092, 1300, 2100, 2275, 3900, 6825, 9100, 27300

The harmonic mean of a set of numbers is defined as the reciprocal of the average of the reciprocals of the numbers. The sum of the 32 reciprocals of the unitary divisors is 32/15 and thus their average is 1/15 which becomes 15 when we consider the reciprocal. Numbers like 27300 comprise OEIS A006086 (permalink):

A006086

Unitary harmonic numbers (those for which the unitary harmonic mean is an integer).

The next unitary harmonic number will occur when I'm 31500 days old which I may or may not be around to celebrate. For posts relating to the harmonic mean see Reciprocals of Primes and Root-Mean-Square And Other Means.

Periodic Unitary Aliquot Sequences.

At first glance, the phrase "periodic unitary aliquot sequences" can sound intimidating so it needs to be broken down into its individual components. Let's start with a definition of aliquot taken from study.com:

An aliquot is a portion or part of a larger whole. An aliquot, or the aliquot part as it is referred to in mathematics, is defined as a positive proper divisor of a number. A divisor refers to a whole number that can be divided evenly into a number.

Using the number associated with my diurnal age today, 27208, the aliquot parts of this number are 1, 2, 4, 8, 19, 38, 76, 152, 179, 358, 716, 1432, 3401, 6802 and 13604. 

The next term to deal with is unitary. The unitary divisors of a number are defined by Wikipedia as follows:

\(a\) is a unitary divisor (or Hall divisor) of a number \(b\) if \(a\) is a divisor of \(b\) and if \(a\) and \(b/a\) are coprime, having no common factor other than 1. Thus, 5 is a unitary divisor of 60, because 5 and 60/5 =12 have only 1 as a common factor, while 6 is a divisor but not a unitary divisor of 60, as 6 and 60/6 = 10 have a common factor other than 1, namely 2. 1 is a unitary divisor of every natural number. 

In the case of 27208, the unitary divisors are 1, 8, 19, 152, 179, 1432, 3401, 27208 but the unitary aliquot divisors are 1, 8, 19, 152, 179, 1432, 3401. See my blog post Unitary Divisors.

Next we need to tackle aliquot sequences. Here is a definition:

Now in the case of 27208, the aliquot sequence terminates in zero. Here is the trajectory (permalink):

27208, 26792, 26668, 21212, 15916, 13316, 9994, 5846, 3274, 1640, 2140, 2396, 1804, 1724, 1300, 1738, 1142, 574, 434, 334, 170, 154, 134, 70, 74, 40, 50, 43, 1, 0

This sequence is terminating and not periodic. However, a unitary aliquot sequence uses the unitary divisors and can behave quite differently. In the case of 27208, the sequence becomes periodic with the following trajectory (permalink):

27208, 5192, 1288, 440, 208, 30, 42, 54, 30

27208 is a member of OEIS A003062:

A003062

Beginnings of periodic unitary aliquot sequences.

The initial members are:

6, 30, 42, 54, 60, 66, 78, 90, 100, 102, 114, 126, 140, 148, 194, 196, 208, 220, 238, 244, 252, 274, 288, 292, 300, 336, 348, 350, 364, 374, 380, 382, 386, 388, 400, 420, 436, 440, 476, 482, 484, 492, 516, 528, 540, 542, 550, 570, 578, 592, 600, 612, 648, 660, 680, 688, 694, 708, 720, 722, 740, 756, 758, 764, 766, 770, 780, 784, 792, 794, 812

Thus periodicity is fairly common for these types of sequences, quite unlike the aliquot sequences which nearly all end in 0. See my post Aliquot Sequences.