The Abundancy Index and Feebly Amicable Numbers

The sum of divisor function, $\sigma(n)$, for a positive integer $n$, is the sum of all the positive divisors including $n$ itself. Looking at the ratio of the sum of divisor function and the number itself, $\frac{\sigma(n)}{n}$, or the abundancy index, we look into its relation to concepts such as perfect numbers, abundant numbers, deficient numbers, and amicable numbers. Through understanding these relations, we will define the concept of feebly amicable numbers also known as harmonious numbers in [1]. This is a generalization of an amicable number with weakened conditions so that the sum of divisors of the two do not need to be equal.

Formally, these are two numbers $m$ and $n$ such that $\frac{n}{\sigma(n)}+\frac{m}{\sigma(m)}=1$. Examples of the first twenty feebly amicable pairs are given for illustration of this concept.

We use the R programming language to produce some abundancy indices and then a list of feebly amicable numbers. This data allows us to ask some questions about feebly amicable numbers that are unknown about amicable numbers.

Our main new results are Theorem 8.1 and Corollary 8.2 which give conditions for when a number can be feebly amicable with another and consequently amicable. The final section has some questions and conjectures that might be of interest to the reader or for future work.

The implications of this research are derived from the historical mathematical workings of famous figures, most notably Euclid, Euler, and Mersenne. Euclid further advanced our understanding of prime numbers by providing the Euclidean algorithm and showing that there are infinitely many prime numbers. Euclid also completed one of the only proofs involving perfect numbers: if $2^{n}-1$ is prime, then $2^{n-1}(2^{n}-1)$ is perfect. Euler continued the idea of perfect numbers by proving that every even perfect number can be expressed in Euclid’s form. This research also makes use of Marin Mersenne’s work on primes, and a Mersenne prime is of the form $2^{n}-1$. [2]

We give some some preliminary definitions that we will rely upon throughout. We are working here with the positive integers (natural numbers). It should also be noted that it is possible to extend all this theory to the negative integers.

Firstly, a divisor is a number that divides into another number without a remainder.Then, a prime is a number that has only the divisors $1$ and itself. Particular prime numbers used in this paper are Mersenne primes which are prime numbers of the form $2^{p-1}(2^{p}-1)$, where $p$ is also prime.

Given a natural number $n$, we can define the canonical representation of $n$ to be $\prod_{i=1}^{r}p_{i}^{a_{i}}$ where the $p_{i}$ are the distinct prime divisors of $n$ and $a_{i}$ their multiplicities.

Two number are coprime (alternatively relatively prime) if they share only $1$ as a divisor.

Then and importantly to this paper, the sum of divisor function $\sigma(n)$ for a positive integer $n$ is defined as the sum of all its divisors (including $n$ itself).

From this, we can categorize natural numbers according to:

• •

  $n$ is called perfect if $\sigma(n)=2n$.

• •

  $n$ is called abundant if $\sigma(n)>2n$.

• •

  $n$ is called deficient if $\sigma(n)<2n$.

This paper generalizes the definition of amicable numbers. To be precise, amicable numbers are two numbers related in such a way that the sum of the proper divisors of each are equal and also equal to the sum of the numbers. That is, $m$,$n$ are amicable if $m+n=\sigma(m)=\sigma(n)$.

Part of this paper is an investigation of the abundancy index of a number. It is defined by $\lambda(n)=\frac{\sigma(n)}{n}$ and, in some sense, measures how divisible a number is.

Multiply-perfect numbers are numbers such that their abundancy index is an integer. Note that perfect numbers by definition have abundancy index $2$.

Finally, two numbers in $\mathbb{N}$ are friendly if they have the same abundancy index. That is, $\frac{\sigma(n)}{n}=\frac{\sigma(m)}{m}$. More generally, friendly numbers form friendly clubs if they all have the same abundancy index.

We now present some preliminary results that give some illustration of the theory and that will be used throughout the rest of the paper. Good references for these and more foundational theory are [3] and [4]. However, there are many good texts on introductory number theory.

The above Proposition 4.1 yields the multiplicative property of the sum of divisors function. That is, if $m$ and $n$ are coprime, then $\sigma(mn)=\sigma(m)\sigma(n)$.

This leads to the following which is Theorem 2.24 of [3]:

We make here a number of remarks regarding values of the abundancy index and how it relates to perfect, abundant, deficient, and friendly numbers.

We will see later that this codomain is not in fact the range, that is, there are values in $\mathbb{Q}\cap(1,\infty)$ that are not abundnacy indices.

We give a proof of the Euclid-Euler Theorem to illustrate the theory:

We also give the following proposition that we will utilize later on:

In order to explore the values of both the sum of divisors function and the abundancy index, we used some code in the R programming language [5] to generate the first 100,000 values of the sum of divisor function and abundancy indices. The following is some code that describes the algorithm:

    sod <- function(x) %This defines the sum of divisors function%
        {s<-0
        for(i in 1:x){if(x%%i==0){s<-s+i}} %This loops through the values 1 through x to see
        which are factors and adds them to the sum if they are%
        return(s)}

    sigma<-c(1:100000) %This defines a vector of length 100,000%

    for(i in 1:100000){sigma[i]<-sod(i)} %This gives a vector of the first 100,000 values of
    the sum of divisors%

    abun<-c(1:100000) %This again defines a vector of length 100,000%

    for(i in 1:100000){abun[i]<-sigma[i]/i} %This gives a vector of the first 100,000 values
    of the abundancy index%

Figure 1 shows a histogram to reflect the data.

By our code we calculated the fraction of abundant numbers in the first $100,000$ numbers to be $0.24799$. This is not within the range given by [6], but it is close. In that paper, they compute that $\alpha=\displaystyle\lim_{n\rightarrow\infty}\frac{A(n)}{n}$ is such that

Here $A(n)$ is the number of abundant numbers less than $n$.

We suggest that computing larger numbers of abundancy indices would give an estimate within the proven bounds. In any case, the histogram illustrates how there are roughly three times more deficient numbers than abundant numbers.

From our data, $\frac{5}{4}$ does not appear as an abundancy index of any $n$ less than $100,000$. First, we prove that $\frac{5}{4}$ does not appear for any natural number, and then, we show a generalization for finding more numbers not in the range.

This can be generalized to Theorem 1 from [7]:

We can see that not all rational numbers greater than $1$ are in the range, but we ask the question: is the range dense in rationals greater than 1? Recall that if $S\subseteq T\subseteq\mathbb{R}$, we say that $S$ is dense in $T$ if for any two numbers in $T$ there exists an element of $S$ in between them.

We refer to [8] and [7] again for the following results:

We give the proofs from [7] of the following results for exposition:

We now proceed to generalize the definition of amicable numbers. We do so by recognizing the following result:

This allows us to formulate the following definition:

Feebly amicable numbers are pairs $m$, $n$ such that

Alternatively, we can define in terms of the abundancy index $\lambda$:

In words, two numbers are feebly amicable if the reciprocals of their abundancy indices sum to 1. Note that feebly amicable pairs are referred to as harmonious pairs in [1].

We also note that amicable pairs have been extended to amicable triples which are three numbers $m$, $n$, and $s$ such that $\sigma(m)=\sigma(n)=\sigma(s)=m+n+s$, and therefore the following is true as well: $\frac{n}{\sigma(n)}+\frac{m}{\sigma(m)}+\frac{s}{\sigma(s)}=1$.

If amicable pairs and triples have been defined such as they have been above, then amicable k-tuples are numbers $n_{1},\ldots,n_{k}$ such that

Given these definitions we can also generalize to feebly amicable triples. These are three numbers $m$, $n$, and $s$ such that

and feebly amicable k-tuples as numbers $n_{1},\ldots,n_{k}$ such that

We now ask the following question: are all integers feebly amicable with some other integer? To see that this is not true, we establish the following:

To see that this is not vacuous note that we have seen that $\frac{5}{4}$ is not an abundancy index, yet the abundancy index of $14182439040$ is known to be $5=\frac{5}{5-4}$ as it is a multiply perfect number of order 5. See OEIS A007539 [9].

A natural corollary of this theorem is the following:

This follows directly from the fact that amicable pairs are feebly amicable pairs. Hence, in particular $14182439040$ has no amicable pair.

Proposition 4.4 showed that the only number with abundancy index $\frac{p+1}{p}$ was $p$ where $p$ was prime. The question arises, is $p$ feebly amicable with any number? Naturally, such a number would need abundancy index $p+1$. That is, $p+1$-perfect. We state this as a result:

It is unknown if there are any coprime amicable pairs [10]. Hence a natural question is whether there are any coprime feebly amicable pairs. The data reveals that there are none less than $1,000$ but that the first coprime feebly amicable pair is $1485$ and $868$. There are another four pairs before $5,000$.

We now generate the first 20 feebly amicable pairs that are not amicable or pairs of perfect numbers. To do so, we implemented the following code:

    for(i in 1:100000){for(j in 1:i){if(1/abun[i]+1/abun[j]==1){print(i) print(j)}}}

These pairs are consistent with those listed in [9] in A253534 and A253535.

This section considers some questions and conjectures that we have encountered in the course of writing this paper.

Question 1 Are there infinitely many coprime feebly amicable pairs?

Given the regularity of coprime pairs of feebly amicable pairs, we conjecture that there are an infinite number.

In [11], Paul Erdős proved that the asymptotic density of amicable integers relative to the positive integers was $0$. That is, the ratio of the number of amicable numbers less than $n$ with $n$ tends to zero as $n$ tends to infinity. This gives rise to the question:

Question 2 What is the density of feebly amicable numbers relative to the positive integers?

Given the number of feebly amicable numbers in the first $5,000$ integers is $310$, then $178$ in the next $5,000$, and $136$ in the third $5,000$, it appears that the density is decreasing and so the asymptotic density is at least less than $0.0272=\frac{136}{15,000}$. Indeed, [1] gives an upper bound and confirms that the asymptotic density is $0$.

The sum of amicable numbers conjecture [12] states that as the largest number in an amicable pair approaches infinity, the percentage of the sums of the amicable pairs divisible by ten approaches $100\%$. We therefore ask the question:

Question 3 As the larger number in a feebly amicable pair approaches infinity, does the percentage of the sums of the pairs divisible by ten approach $100\%$?

Our data tells us that in the first $5000$ numbers there are 11 feebly amicable pairs that sum to a multiple of 10. There are then 8 in the next $5000$ and 4 between $10000$ and $15000$. As a sequence of fractions of the number of feebly amicable pairs this is: $0.035,0.045,0.029$. This does not give us any noticeable trend and once again much higher values would be required to make a strong conjecture other than to say it does not appear that the percentage tends towards 100$\%$ as in the amicable case. Computing further is certainly possible, but it becomes much more computationally expensive to compute sum of divisors and hence abundancy indices. [13] provides an algorithm to compute in $O(n^{\frac{1}{3}})$ time, but our code was much more crude.