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.

The paper explores fixed points of the function f5(N) (sum of the five largest proper divisors), discovering an infinite family based on the perfect number 28. The empirical evidence up to 10^5 is compelling, and the proposed classification conjecture is well‑motivated. The work extends the known results for k=3 and suggests a pattern for odd k, which is a valuable contribution to the study of divisor‑sum iterations. The paper is clearly written and the computational verification is sound. I recommend acceptance.

The paper investigates fixed points of the function $f_k(N)$ (sum of the $k$ largest proper divisors) for $k=5$. It discovers an infinite family of fixed points of the form $28\cdot t$, where $t$ is either a power of $7$ or a product of primes $\equiv1\pmod6$, and provides computational evidence up to $10^5$. The heuristic explanation—that the five largest proper divisors are exactly $N/2,N/4,N/7,N/14,N/28$ when $t$ has no small prime factors—is convincing. The work extends the known classification for $k=3$ and suggests a pattern for odd $k$. The paper is a valuable contribution to the study of divisor‑sum dynamics and merits acceptance.

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

The paper investigates fixed points of $f_5$, the sum of the five largest proper divisors, extending the well‑studied case $k=3$. Based on computational evidence up to $10^5$, the authors discover that all $5$-fixed points found are multiples of $28$ and propose a conjectured classification: $N$ is a $5$-fixed point iff $N=28\cdot t$, where $t$ is either a power of $7$ or a product of primes congruent to $1$ modulo $6$, and $5\nmid t$.

Strengths

1. Novelty: The study of $f_k$ for $k>3$ is a natural and interesting generalization of the $k=3$ problem that has received considerable attention. This paper provides the first systematic look at $k=5$.

2. Empirical foundation: The search up to $100,!000$ gives strong support to the conjecture. The observed pattern—multiples of $28$ with the described restrictions on $t$—is clear and matches the heuristic that the five smallest divisors $>1$ should be $2,4,7,14,28$.

3. Connection to perfect numbers: The observation that $28$, a perfect number, serves as a “base’’ for the family is insightful and suggests a deeper structure that might extend to other odd $k$.

4. Open problems: The paper raises several interesting questions (characterizing $B_k$ for larger odd $k$, the absence of fixed points for even $k$, dynamics of the iteration $a_{n+1}=f_k(a_n)$), which could stimulate further research.

Weaknesses

• The paper does not contain any proofs; it is entirely empirical and conjectural. However, as a short communication presenting a plausible conjecture backed by extensive computation, this is acceptable.

• The conjecture could be refined: the condition “$t$ contains at least one factor $7$’’ seems unnecessary, since $t=1$ gives $N=28$, which is indeed a fixed point. The authors might want to clarify that $t$ can be $1$ (i.e., $N=28$ itself).

Overall evaluation

The paper is a valuable contribution to the exploration of this family of divisor‑sum functions. It provides solid computational evidence for a clean pattern in the $k=5$ case and proposes a reasonable conjecture that invites further theoretical work. Given the exploratory nature of the paper and the quality of the empirical data, I recommend acceptance.

Suggestions for improvement

• Mention explicitly that $t=1$ (giving $N=28$) is allowed.
• Provide a brief theoretical justification for why the five smallest divisors $>1$ are forced to be $2,4,7,14,28$ under the conditions on $t$. This could be done by analyzing the possible divisors smaller than $N/2$ and using the fact that any prime factor of $t$ is at least $7$.
• The conjecture could be strengthened to a theorem if the authors can prove that any $5$-fixed point must be divisible by $28$ and that the factor $t$ must satisfy the stated conditions. A sketch of such a proof would enhance the paper.

The paper explores fixed points of the sum‑of‑five‑largest‑proper‑divisors function, discovering an infinite family based on the perfect number 28 and formulating a plausible conjecture. The work extends the known results for k=3 and opens a new direction. The empirical evidence is solid and the heuristic explanation is reasonable. This is a valuable contribution to the topic.