A number $n$ which is the sum of all numbers $k$ with $\sigma(k)=n$?

For a positive integer $n$, let us define a set

$$A_n = \{ k\in\mathbb{N} \mid \sigma(k) = n \}$$

where $\sigma$ is the divisor-sum function (a well-known multiplicative number-theoretic function). Clearly $A_n \subseteq \{ 1,2,3,\ldots,n\}$ (since $\sigma(k)\ge k$ for all $k$).

For example

$$\begin{align} A_{119} & = \varnothing \\ A_{120} & = \{ 54, 56, 87, 95 \} \\ A_{121} & = \{ 81 \}. \end{align}$$

Now denote by $\Sigma_n$ the sum of the members of $A_n$, so $\Sigma_n = \sum_{k\in A_n}k$, so (continuing the example)

$$\begin{align} \Sigma_{119} & = 0 \\ \Sigma_{120} & = 292 \\ \Sigma_{121} & = 81. \end{align}$$

Note that $\Sigma_{119}<119$ and $\Sigma_{121}<121$, and on the other hand $\Sigma_{120}>120$.

This splits the natural numbers $n$ into three classes, according to whether $\Sigma_n<n$, $\Sigma_n=n$, or $\Sigma_n>n$. I find a lot of numbers in the first and the last of these classes. However, the only number with $\Sigma_n = n$ that I have found is the trivial case $n=1$.

Are there any numbers $n>1$ with $\Sigma_n = n$?

PS! I am planning on submitting new sequences to OEIS if people find this partition of $\mathbb{N}$ interesting.

Here are some statistics for all $n$ in $\left[ 1, 60000 \right]$:

$$\begin{array}{|r|r|r|r|} n \pmod{6} & \Sigma_n<n & \Sigma_n=n & \Sigma_n>n \\ \hline +1 \pmod{6} & 9993 & 1 & 6 \\ +2 \pmod{6} & 9020 & 0 & 980 \\ 3 \pmod{6} & 9992 & 0 & 8 \\ -2 \pmod{6} & 9415 & 0 & 585 \\ -1 \pmod{6} & 10000 & 0 & 0 \\ 0 \pmod{6} & 5958 & 0 & 4042 \\ \hline \mathrm{total} & 54378 & 1 & 5621 \\ \end{array}$$

Update: I searched a bit further, $\left[ 1,\quad 300\cdot 10^6 \right]$:

$$\begin{array}{|r|r|r|r|} n \pmod{6} & \Sigma_n<n & \Sigma_n=n & \Sigma_n>n \\ \hline +1 \pmod{6} & 49999688 & 1 & 311 \\ +2 \pmod{6} & 47797853 & 0 & 2202147 \\ 3 \pmod{6} & 49999279 & 0 & 721 \\ -2 \pmod{6} & 47343370 & 0 & 2656630 \\ -1 \pmod{6} & 49999985 & 0 & 15 \\ 0 \pmod{6} & 36529965 & 0 & 13470035 \\ \hline \mathrm{total} & 281670140 & 1 & 18329859 \\ \end{array}$$

The first $n$ with $n \equiv -1 \pmod{6}$ so that $\Sigma_n>n$ is $86831$. We have $A_{86831} = \{ 38416, 60025 \}$.

A value for which $\Sigma_n=n$ corresponds to an amicable tuple which comprises all numbers with that $\sigma$ value, i.e. $A_n$ is amicable. We could call that a total amicable tuple. This question then becomes if any total amicable tuples other than $\{ 1 \}$ exist.

I have now created A258913 in OEIS which gives what is called $\Sigma_n$ above. According to comment by Giovanni Resta there, any new $n$ with $\Sigma_n=n$ will exceed $2.5\cdot 10^{10}$.

Solution 1:

*Not an answer, but too long for a comment.

As pointed out in the comments, an obvious family of near misses (precisely, a family where $|\Sigma_n - n|\in O(1)$) comes from the primes: assuming $A_n = \{ p \}$, then $n=\sigma(p)=p+1$, and $\Sigma_n = p = n-1$. Not every prime $p$ works, because $A_{p+1}$ may contain other elements besides $p$, but infinitely many seem to.

A less-obvious family of near misses comes from pairs of primes $(p,q)$ for which $q=4p+3$. Assuming $A_n = \{ 12p, 4q \}$, then $$ n=\sigma(12p)=\sigma(2^2 \cdot 3 \cdot p)=(1+2+4)(1+3)(1+p)=28p+28,\\ n=\sigma(4q)=\sigma(2^2 q)=(1+2+4)(1+q)=7(1+q)=28p+28, $$ and $$ \Sigma_n=12p + 4q=12p+4(4p+3)=28p+12=n-16. $$ Again, this doesn't always work, because $A_{28p+28}$ may contain elements other than $12p$ and $4q$, but it appears to happen infinitely often.

Just inspecting the data doesn't turn up any other obvious families like this ($-1$ and $-16$ are by far the most recurrent values of $\Sigma_n - n$), but I wonder if they exist and are simply sparse. Can this be generalized? Is there another near-miss family for which $|A_n|=2$, or a near-miss family with $|A_n|=3$, for instance?

Solution 2:

I have an idea that may be a partial answer. My hunch is that it is only true for $n=1$.

Let's rephrase your question: What you're really asking is "Can $k_1 + k_2 + \cdots + k_m = \sigma(k_1)$?" where $m = |A_{n}|$ and such that $\sigma(k_i) = n$ for all $i \leq m$. You may already know the following terminology, but for those who don't:

Consider the $\bf{abundance}$ of $k$, defined and denoted as $I(k) = \displaystyle \frac{\sigma(k)}{k}$.

Observe that $\displaystyle \frac{\sigma(k)}{k} > 1$ when $k > 1$.

If $I(k) > 2$, call $k$ $\bf{abundant}$.

If $I(k) = 2$, call $k$ $\bf{perfect}$.

If $I(k) < 2$, call $k$ $\bf{deficient}$.

Here's a start with cases:

1) WLOG, assume $k_1$ is deficient. Thus $\displaystyle \frac{k_2}{k_1} + \cdots + \frac{k_m}{k_1} < 1$. Thus, all $k_i$ are more abundant than $k_1$. WLOG, assume $k_2 < k_1$. Further, assume $k_2$ is still deficient. Then $\displaystyle \frac{k_1}{k_2} + \frac{k_3}{k_2} + \cdots + \frac{k_m}{k_2} < 1$, but this is a contradiction since $\displaystyle \frac{k_1}{k_2} > 1$. Thus, if such $A_n$ is to exist, it cannot have more than one deficient element. In fact, it's similar to show that such $A_n$ cannot have together a deficient element and a perfect element. So the other elements must be abundant.

2) WLOG, assume $k_1$ is perfect. Then $\displaystyle \frac{k_2}{k_1} + \cdots + \frac{k_m}{k_1} = 1$, and, for similar reasoning as before, each $k_i$ must be abundant. So, if such $A_n$ is to exist, it can have at most one perfect element, and the rest must be abundant.

3) WLOG, assume all $k_i$ are abundant. Thus $\sigma(k_i) > 2k_i$ for all $i$. This implies each $k_i < \displaystyle\frac{n}{2}$. Thus we know from this fact that $A_n$ needs at least $3$ elements.

I'm currently still working on this case to (hopefully) show a contradiction. I'm working with many different methods, so it might be a few days.

Overall, we've shown that if such an $A_{n}$ exists, it needs to have at least three primarily abundant elements; "primarily" meaning "all but one."