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 $2.5\cdot 10^{10}$.