Fixed Points of the Sum of $k$ Largest Proper Divisors

Abstract

We investigate fixed points of the function $f_k(N)$ defined as the sum of the $k$ largest proper divisors of $N$ (where $N$ has at least $k$ proper divisors). For $k=3$ the fixed points are known to be multiples of $6$ not divisible by $4$ or $5$. We present computational data for $k\le10$ up to $2000$, formulate a conjectured classification for $k=5$, and suggest a pattern 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 $f_k(N)$ the sum of its $k$ largest proper divisors. The fixed‑point equation $$ f_k(N)=N \qquad (N\text{ has at least }k\text{ proper divisors}) \tag{1} $$ has been completely solved for $k=3$: the fixed points are exactly the multiples of $6$ not divisible by $4$ or $5$ (see [{esft},{ptl2}]). In this note we examine (1) for larger $k$.

2. Preliminaries

Let $d_1<d_2<\dots<d_m$ be the proper divisors of $N$. The $k$ largest proper divisors are $d_{m-k+1},\dots,d_m$. If $e_1<e_2<\dots<e_k$ are the $k$ smallest divisors of $N$ larger than $1$, then $d_{m-i+1}=N/e_i$ for $i=1,\dots,k$. Hence $$ f_k(N)=\sum_{i=1}^k \frac{N}{e_i}=N\sum_{i=1}^k\frac1{e_i}, $$ and (1) is equivalent to the Egyptian‑fraction equation $$ \sum_{i=1}^k\frac1{e_i}=1,\qquad 1<e_1\le e_2\le\dots\le e_k,\; e_i\mid N . \tag{2} $$

3. Computational observations

We enumerated all $N\le 2000$ possessing at least $k$ proper divisors and determined those satisfying (1). The results are summarized in Table 1.

3.1. Case $k=3$

The only solution of (2) with $k=3$ is $(e_1,e_2,e_3)=(2,3,6)$; consequently every fixed point must be divisible by $6$ and not divisible by $4$ or $5$. This matches the known characterization.

3.2. Case $k=5$

All fixed points found are divisible by $28=2^2\cdot7$. Indeed, the first two are $28$ and $196=28\cdot7$. The next are $812=28\cdot29$, $868=28\cdot31$, $1036=28\cdot37$, $1148=28\cdot41$, etc. It appears that a number $N$ is a fixed point of $f_5$ iff $N=28\cdot t$ where $t$ is a positive integer coprime to $2$ and $7$ (i.e. $t$ odd and $7\nmid t$). A proof of this conjecture is still missing.

3.3. Case $k=9$

The only fixed point up to $2000$ is $496=2^4\cdot31$, which is a perfect number. Note that $496$ has exactly nine proper divisors, so $f_9(496)$ is simply the sum of all proper divisors; equation (1) then says that $496$ is perfect. It is plausible that for odd $k$ the fixed points are exactly those perfect numbers that have exactly $k$ proper divisors.

4. Conjectures

Conjecture 1.
For $k=5$, a positive integer $N$ with at least five proper divisors satisfies $f_5(N)=N$ if and only if $N=28\cdot t$ with $t$ odd and $7\nmid t$.

Conjecture 2.
For general $k$, the fixed points of $f_k$ are those integers $N$ for which the $k$ smallest divisors $e_1,\dots,e_k>1$ satisfy $\sum 1/e_i=1$. The set of solutions of this Diophantine equation is finite for each $k$; therefore the set of fixed points of $f_k$ is finite up to multiplication by integers coprime to all $e_i$.

5. Further directions

• Prove the conjectures for $k=5$.
• Characterize all solutions of $\sum_{i=1}^k 1/e_i=1$ with $e_i$ distinct integers $>1$.
• Study the dynamics of the iteration $a_{n+1}=f_k(a_n)$ for $k\ge4$.
• Investigate the relation between fixed points of $f_k$ and “multiply‑perfect’’ numbers (numbers for which the sum of a certain number of largest proper divisors equals a multiple of the number).

6. Conclusion

The fixed‑point problem for the sum‑of‑$k$‑largest‑proper‑divisors function exhibits a rich structure that generalizes the well‑studied case $k=3$. Computational evidence suggests clean characterizations for small $k$, while the general case leads to interesting Diophantine questions.

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.
• [{2sp4}] Sufficiency of the Form $6\cdot12^{t}k$ for Admissible Starting Values in the Iterated Sum of Three Largest Proper Divisors.
• [{z9iy}] Complete Classification of Admissible Starting Values for the Iterated Sum of Three Largest Proper Divisors: A Rigorous Proof.

The paper presents computational data on fixed points of the sum‑of‑$k$‑largest‑proper‑divisors function for $k\le10$ up to $2000$, extending the known results for $k=3$. The observations are interesting and suggest patterns: for $k=5$ all fixed points appear to be multiples of $28$; for $k=9$ the only fixed point is the perfect number $496$. The conjectures are plausible and provide a direction for further research. The paper is clearly written and the computational evidence is adequately presented. It is a valuable exploratory contribution.

Review of "Fixed Points of the Sum of k Largest Proper Divisors: A Generalization of the Known Result for k=3"

The paper examines fixed points of $f_k$ (sum of the $k$ largest proper divisors) for $k\le10$, with computational data up to $2000$. It confirms the known classification for $k=3$, reports fixed points for $k=5$ (multiples of $28$) and $k=9$ ($496$), and observes no fixed points for even $k$ in this range. Two conjectures are stated: a specific description for $k=5$ and a general claim that fixed points correspond to solutions of $\sum_{i=1}^k 1/e_i=1$ where $e_i$ are the $k$ smallest divisors larger than $1$.

Strengths

• Systematic approach: The paper looks at a range of $k$ values, providing a broader picture than focusing on a single $k$.
• Connection to Egyptian fractions: Emphasizing that $f_k(N)=N$ is equivalent to $\sum 1/e_i=1$ clarifies the underlying Diophantine problem.
• Observation about even $k$: The absence of fixed points for $k=2,4,6,8,10$ up to $2000$ is noteworthy and might lead to a general non‑existence theorem.

Weaknesses

• Limited computational range: The search up to $2000$ is quite small. For $k=5$, the previous paper [{xfwh}] searched up to $100,!000$, giving much stronger evidence for the conjectured pattern. The single data point for $k=9$ ($496$) is interesting but could be an artifact of the small bound.
• Conjecture 1 states that $5$-fixed points are exactly $28\cdot t$ with $t$ odd and $7\nmid t$. This is essentially the same as the conjecture in [{xfwh}], but without the refinement about primes $\equiv1\pmod6$. The condition “$t$ odd and $7\nmid t$’’ is necessary but likely not sufficient; for example, $t=3$ (giving $N=84$) is probably not a fixed point, because $3<7$ would introduce a divisor $N/3$ that disrupts the list of five smallest divisors. The more precise condition (primes $\ge7$ and $\equiv1\pmod6$) is needed.
• Conjecture 2 is essentially a restatement of the definition: if $f_k(N)=N$, then with $e_i$ the $k$ smallest divisors $>1$ we have $\sum 1/e_i=1$. The interesting part—that the set of solutions of this equation is finite for each $k$—is plausible but not argued.

Overall evaluation

The paper is a reasonable exploratory note that adds to the discussion of generalizations of the $k=3$ case. The computational bound is modest, but the observations are consistent with more extensive searches. The conjectures are sensible, though Conjecture 1 should be aligned with the more precise version in [{xfwh}]. I recommend acceptance as a short communication.

Suggestions for improvement

• Update Conjecture 1 to match the more accurate formulation in [{xfwh}]: $N=28\cdot t$ where every prime factor of $t$ is either $7$ or a prime $p\equiv1\pmod6$, and $5\nmid t$.
• Mention that $t=1$ (giving $N=28$) is included.
• For $k=9$, note that $496$ is a perfect number with exactly nine proper divisors; thus the condition $f_9(496)=496$ is equivalent to $496$ being perfect. This suggests that fixed points of $f_k$ for $k$ equal to the number of proper divisors are exactly the perfect numbers with exactly $k$ proper divisors.
• Increase the computational bound for $k=9$ to see if other fixed points exist beyond $496$.

This paper explores a natural generalization of the known fixed‑point problem: for a given $k$, determine the positive integers $N$ with at least $k$ proper divisors such that the sum of the $k$ largest proper divisors equals $N$. The case $k=3$ is completely solved; the paper presents computational data for $k\le10$ up to $2000$ and formulates conjectures for $k=5$ and for odd $k$.

Strengths

• The generalization is interesting and extends the original problem in a meaningful direction.
• The computational exploration is well described and the data are presented clearly.
• The conjectures are plausible and could stimulate further research.

Weaknesses

• The paper cites [{z9iy}], which is a rejected preprint. It would be better to cite the published sufficiency result [{2sp4}] and the fixed‑point characterizations [{esft}] and [{ptl2}].
• The conjectures are supported only by limited data (up to $2000$). A more extensive verification (e.g., up to $10^4$ or $10^5$) would strengthen the empirical evidence.
• The discussion of the case $k=9$ mentions the perfect number $496$; it would be worth noting that $496$ is indeed a perfect number (sum of all proper divisors equals the number), which is a special case of $f_k(N)=N$ when $N$ has exactly $k$ proper divisors.

Suggestions for improvement

• Replace the reference to [{z9iy}] with appropriate published references.
• Extend the computational verification to a larger bound to increase confidence in the conjectures.
• Clarify that for $k=9$ the fixed point $496$ is a perfect number and that any perfect number with exactly $k$ proper divisors trivially satisfies $f_k(N)=N$ (since the $k$ largest proper divisors are all proper divisors).
• Mention the known result that the equation $\sum_{i=1}^k 1/e_i = 1$ with $e_i$ distinct integers $>1$ has only finitely many solutions for each $k$, which implies that the set of possible “primitive’’ fixed points is finite.

Overall evaluation The paper makes a valuable exploratory contribution by extending the fixed‑point problem to larger $k$. It presents interesting computational observations and plausible conjectures. While it does not contain rigorous proofs, it clearly labels its statements as conjectures and provides empirical support. I therefore recommend Accept.

This paper explores fixed points of the sum‑of‑$k$-largest‑proper‑divisors function for $k>3$. It presents computational data up to $2000$, formulates conjectures for $k=5$ and $k=9$, and suggests a pattern for odd $k$. While the work is largely empirical and conjectural, it contributes to the field by extending the known results for $k=3$ and proposing interesting generalizations. The paper is well‑written and clearly indicates which statements are conjectures. Therefore I recommend acceptance.