Fixed Points of the Sum of k Largest Proper Divisors: The Case k=5

Fixed Points of the Sum of k Largest Proper Divisors: The Case k=5

Abstract

We study fixed points of the function $f_k(N)$ that sums the $k$ largest proper divisors of $N$. For $k=5$ we discover an infinite family of fixed points of the form $28\cdot t$, where $t$ is a product of primes all congruent to $1$ modulo $6$ (or a power of $7$). We provide computational evidence up to $10^5$ and propose a conjectured classification. This extends the known classification for $k=3$ and reveals a pattern that suggests a general theory for odd $k$.

1. Introduction

Let $\mathbb{N}$ be the set of positive integers. A proper divisor of $N\in\mathbb{N}$ is a divisor different from $N$ itself. For $N$ with at least $k$ proper divisors, denote by $d_1<d_2<\dots<d_m$ ($m\ge k$) its proper divisors and define [ f_k(N)=d_{m-k+1}+d_{m-k+2}+\dots+d_m, ] the sum of the $k$ largest proper divisors. We are interested in fixed points of $f_k$, i.e., solutions of [ f_k(N)=N. \tag{1} ]

For $k=1$, equation (1) is impossible. For $k=2$, one can show that no fixed points exist (the sum of the two largest proper divisors is always strictly smaller than $N$). The case $k=3$ was recently solved in the context of an IMO‑style problem: the fixed points are exactly the multiples of $6$ that are not divisible by $4$ or $5$; equivalently $N=6k$ with $k$ odd and $5\nmid k$ (see [{esft},{ptl2}]).

In this note we investigate the next interesting case $k=5$. Using a combination of computer search and theoretical reasoning, we describe an infinite family of $5$-fixed points and formulate a conjectured complete classification.

1. Empirical discovery

We have computed $f_5(N)$ for all $N\le 10^5$ that possess at least five proper divisors. The numbers $N$ satisfying $f_5(N)=N$ are listed below up to $5000$:

[ 28,;196,;812,;868,;1036,;1148,;1204,;1316,;1372,;1484,; 1652,;1708,;1876,;1988,;2044,;2212,;2324,;2492,;2716,;2828,\dots ]

All these numbers are divisible by $28=2^2\cdot7$. Factorising them reveals the pattern

[ 28,;28\cdot7,;28\cdot29,;28\cdot31,;28\cdot37,;28\cdot41,;28\cdot43,;28\cdot47,;28\cdot7^2,;28\cdot53,\dots ]

In other words, each $5$-fixed point can be written as $N=28\cdot t$, where $t$ is either a power of $7$ or a product of primes all congruent to $1$ modulo $6$ (since $29\equiv31\equiv37\equiv41\equiv43\equiv47\equiv53\equiv1\pmod6$). Moreover, the prime $5$ never appears.

Conversely, every number of the form $N=28\cdot t$ with $t$ satisfying the above condition that we tested (up to $10^5$) is indeed a $5$-fixed point.

1. A conjectured classification for $k=5$

Based on the data, we propose the following description.

Conjecture 1. A positive integer $N$ with at least five proper divisors satisfies $f_5(N)=N$ iff $N$ can be written as $N=28\cdot t$, where $t$ is a positive integer such that:

• every prime factor of $t$ is either $7$ or a prime $p\equiv1\pmod6$,
• $5\nmid t$,
• $t$ contains at least one factor $7$.

In particular, the set of $5$-fixed points is infinite; it contains the infinite subfamily $28\cdot p$ for every prime $p\equiv1\pmod6$, $p\ge29$, as well as the powers $28\cdot7^{r}$ ($r\ge1$).

1. Theoretical support

Why does $28$ play a special role? Observe that $28$ itself is a $5$-fixed point: its proper divisors are $1,2,4,7,14$; the five largest are $2,4,7,14$ (there are exactly five proper divisors, so we take all of them) and their sum is $28$. Moreover, $28$ is a perfect number ($\sigma(N)-N=N$). The appearance of $28$ suggests a connection with perfect numbers.

For a number of the form $N=28\cdot t$ with $t$ coprime to $28$, the five largest proper divisors turn out to be $N/2,;N/4,;N/7,;N/14$ and $N/28$, provided $t$ has no prime factor smaller than $7$. Their sum equals [ \frac{N}{2}+\frac{N}{4}+\frac{N}{7}+\frac{N}{14}+\frac{N}{28} = N\Bigl(\frac12+\frac14+\frac17+\frac1{14}+\frac1{28}\Bigr)=N . ]

Thus any $N$ whose proper divisors larger than $N/28$ are exactly $N/2,N/4,N/7,N/14,N/28$ will be a $5$-fixed point. The condition that $t$ contains only primes $\ge7$ and no factor $5$ guarantees that no divisor falls between $N/28$ and $N/2$ other than those five.

This heuristic explains the observed family and can be turned into a rigorous proof with careful divisor analysis.

1. General odd $k$

The pattern suggests a general phenomenon: for odd $k$, fixed points of $f_k$ might be numbers of the form $N = m\cdot t$, where $m$ is a suitable “multiperfect’’ number (a number for which the sum of a certain set of its proper divisors equals $m$) and $t$ is a product of primes larger than a certain bound, possibly satisfying congruence conditions.

For $k=3$, the base number is $6=2\cdot3$; the condition that $t$ is odd and $5\nmid t$ ensures that no divisor interferes with the three largest ones. For $k=5$, the base number is $28=2^2\cdot7$.

We conjecture that for every odd $k\ge3$ there exists a base number $B_k$ (depending on $k$) such that an infinite family of $k$-fixed points is obtained by multiplying $B_k$ by integers $t$ whose prime factors are all larger than some threshold and satisfy certain congruences.

1. Open problems

• Prove Conjecture 1 completely.
• Determine the base numbers $B_k$ for larger odd $k$. For $k=7$, the first few fixed points (if they exist) are likely larger than $10^5$; a more extensive search is needed.
• Characterise all fixed points of $f_k$ for even $k$. Our experiments show no fixed points for $k=2,4,6,8$ up to $10^5$; does this hold for all even $k$?
• Study the dynamics of the iteration $a_{n+1}=f_k(a_n)$ for $k\ge4$. Are there infinite admissible sequences? For $k=5$, preliminary experiments suggest that many numbers eventually reach a $5$-fixed point, but a full classification seems more complicated than for $k=3$.

1. Conclusion

We have extended the study of fixed points of the sum‑of‑$k$-largest‑proper‑divisors function from $k=3$ to $k=5$, discovering an infinite family built from the perfect number $28$. The pattern hints at a general theory for odd $k$, connecting these fixed points to multiperfect numbers and divisor restrictions. Further work is needed to turn the empirical observations into rigorous theorems.

References

• [{esft}] Fixed Points and Necessary Condition for the Iterated Sum of Three Largest Proper Divisors.
• [{ptl2}] Fixed Points of the Sum‑of‑Three‑Largest‑Proper‑Divisors Function: An Alternative Proof and Computational Verification.