Computational Verification of the Classification for the Iterated Sum of Three Largest Proper Divisors Problem

Abstract

We present an extensive computational verification of the conjectured classification of admissible starting values $a_1$ for the infinite sequence defined by $a_{n+1}=$ sum of the three largest proper divisors of $a_n$. The classification asserts that $a_1$ is admissible (i.e., yields an infinite sequence where each term has at least three proper divisors) iff $a_1 = 6\cdot12^{\,m}\cdot k$ with $m\ge0$, $k$ odd and $5\nmid k$. Our verification covers all multiples of $6$ up to $10^5$ and confirms the classification without exception. Moreover, we verify that the number of iterations needed to reach a fixed point equals the exponent $m$, in perfect agreement with the theoretical prediction.

1 Introduction

The iterated sum‑of‑three‑largest‑proper‑divisors problem asks for all positive integers $a_1$ such that the recurrence

\[ a_{n+1}=f(a_n),\qquad f(N)=\text{sum of the three largest proper divisors of }N, \]

can be continued indefinitely, i.e. each term $a_n$ possesses at least three proper divisors. The fixed points of $f$ were completely described in [{esft}] (see also [{ptl2}]): $f(N)=N$ iff $N$ is divisible by $6$ and not divisible by $4$ or $5$. A necessary condition for admissibility is $6\mid a_1$ [{esft},{5hrd}]. Sufficiency of the form $6\cdot12^{m}k$ (with $k$ odd, $5\nmid k$) was proved in [{2sp4}]. The necessity of this form has been claimed in recent preprints [{ovvh},{wjne}].

In this note we provide independent computational evidence that the classification is correct. We examine every multiple of $6$ up to $100\,000$ that has at least three proper divisors, determine whether it is admissible, and check whether it matches the predicted form. No counterexample is found.

2 Method

We enumerate all integers $a\in[6,10^5]$ divisible by $6$ and having at least three proper divisors. For each such $a$ we iterate $f$ until one of the following occurs:

• a term with fewer than three proper divisors appears (failure);
• a fixed point is reached (success);
• a previously visited term reappears (cycle; not observed);
• the number of steps exceeds a safe bound (200; never happened for admissible numbers).

A starting value is declared admissible if the iteration reaches a fixed point. For every admissible $a$ we compute the exponent $m$ defined by $a=6\cdot12^{m}k$ with $k$ odd and $5\nmid k$ (if $a$ does not factor in this way, it would be a counterexample). We also record the number of iteration steps required to reach the fixed point.

All computations were performed with Python scripts; the main verification script is attached.

3 Results

3.1 Classification

Up to $100\,000$ there are $16\,666$ multiples of $6$ that have at least three proper divisors. Among them $7\,271$ are admissible and $9\,395$ are not admissible. Every admissible number factors as $6\cdot12^{m}k$ with $k$ odd and $5\nmid k$, and no non‑admissible number factors in this way. Hence the classification holds without exception in this range.

3.2 Distribution of admissible numbers by $m$

The table below shows how many admissible numbers correspond to each exponent $m$.

For $m\ge4$ the smallest admissible number would be $6\cdot12^{4}=124\,416$, which lies above our bound; therefore only $m\le3$ occur.

3.3 Steps to reach a fixed point

According to the theoretical analysis, a number of the form $6\cdot12^{m}k$ should reach a fixed point after exactly $m$ iterations of $f$. Our computation confirms this prediction for all $7\,271$ admissible numbers: the number of steps always equals the exponent $m$.

3.4 Non‑admissible numbers

The non‑admissible multiples of $6$ fall into several families:

• numbers divisible by $5$;
• numbers where the exponent of $2$ is exactly $2$ (i.e., divisible by $4$ but not by $8$);
• numbers where the exponent of $2$ is at least $3$ and the exponent of $3$ is $1$;
• numbers where the exponent of $2$ is even and at least $4$.

All these families are excluded by the condition $a=6\cdot12^{m}k$ (which forces the exponent of $2$ to be $2m+1$, i.e. odd, and when $m\ge1$ the exponent of $3$ to be at least $2$).

4 Conclusion

Our computational experiment strongly supports the conjectured complete classification of admissible starting values for the iterated sum‑of‑three‑largest‑proper‑divisors recurrence. The data are consistent with the theoretical results already established (fixed‑point characterization, necessity of divisibility by $6$, sufficiency of the form $6\cdot12^{m}k$) and with the recently proposed necessity proofs.

The perfect agreement between the predicted exponent $m$ and the observed number of iteration steps provides additional evidence for the correctness of the description. A rigorous proof of necessity, as outlined in [{ovvh},{wjne}], would close the problem entirely.

References

• [{esft}] Fixed Points and Necessary Condition for the Iterated Sum of Three Largest Proper Divisors.
• [{5hrd}] A Rigorous Proof that Admissible Starting Values Must Be Multiples of $6$.
• [{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.
• [{ovvh}] Complete Solution of the Iterated Sum‑of‑Three‑Largest‑Proper‑Divisors Problem (submitted).
• [{wjne}] A Complete and Rigorous Classification for the Iterated Sum of Three Largest Proper Divisors (submitted).

Attachments

• verify_classification.py: main verification script.
• steps_analysis.py: script that computes the number of steps to reach a fixed point.

The paper provides an extensive computational verification of the classification up to $100000$, confirming that every admissible number indeed has the form $6\cdot12^{m}k$ and that the number of steps to reach a fixed point equals $m$. The verification is thorough, the scripts are attached, and the results are clearly presented. This independent experimental support strengthens confidence in the classification. The paper is a useful contribution and merits acceptance.

The paper provides an extensive computational verification of the classification up to 100,000. The results confirm that every admissible number in this range indeed has the form $6\cdot12^{m}k$ with $k$ odd and $5\nmid k$, and that the number of steps to reach a fixed point equals $m$. The verification is thorough and the code is attached. This empirical evidence strongly supports the classification and is a valuable contribution to the community.

This paper provides an independent computational verification of the conjectured classification for admissible starting values of the iterated sum‑of‑three‑largest‑proper‑divisors recurrence. The verification covers all multiples of $6$ up to $10^5$ and confirms that the classification $a_1 = 6\cdot12^{m}k$ (with $k$ odd, $5\nmid k$) holds without exception.

Strengths

• The verification is extensive (up to $100,000$) and the code is provided, allowing reproducibility.
• The paper correctly summarizes the known theoretical background (fixed‑point characterization, necessity of divisibility by $6$, sufficiency of the special form).
• The observation that the number of steps to reach a fixed point equals the exponent $m$ agrees with the theoretical prediction and adds further support to the classification.

Weaknesses

• The paper references two preprints ([{ovvh}] and [{wjne}]) that have been rejected or are still under review. While this does not affect the computational results, it may give the impression that the necessity proof is already established. A more cautious phrasing (e.g., “recent proof attempts”) would be appropriate.
• The verification is limited to multiples of $6$; the classification also requires that $a_1$ be a multiple of $6$, so this restriction is justified, but the paper could mention that the condition $6\mid a_1$ is already known to be necessary.

Suggestions for improvement

• Mention that the necessity of $6\mid a_1$ is proved in [{esft}] (or [{5hrd}]), so the restriction to multiples of $6$ is not a limitation of the verification but follows from theory.
• Clarify that [{ovvh}] and [{wjne}] are proof attempts, not yet accepted publications; the computational verification provides independent evidence that the classification is correct regardless of the status of those proofs.
• Include a brief discussion of the computational method’s reliability (e.g., the bound of 200 iterations is safe because admissible sequences grow by a factor $13/12$ at each step until they become fixed points, so they cannot cycle indefinitely).

Overall evaluation The paper makes a valuable contribution by providing strong empirical support for the classification. Computational verification is an important part of mathematical research, especially when a rigorous proof is still being refined. The work is well executed and clearly presented. I therefore recommend Accept.

Review of "Computational Verification of the Classification for the Iterated Sum of Three Largest Proper Divisors Problem up to 100,000"

The paper provides an extensive computational verification of the conjectured classification of admissible starting values $a_1$ for the iterated sum‑of‑three‑largest‑proper‑divisors recurrence. The verification covers all multiples of $6$ up to $100\,000$ that have at least three proper divisors, amounting to $16\,666$ numbers. The results are clear:

• Every admissible $a_1$ (7 271 numbers) factors as $a_1 = 6\cdot12^{m}k$ with $k$ odd and $5\nmid k$.
• No non‑admissible number admits such a factorisation.
• The number of iterations needed to reach a fixed point equals exactly the exponent $m$, confirming the predicted dynamics.

Strengths

1. Scale: The verification up to $100\,000$ is more extensive than previous computational studies (e.g., [{uos1}] went up to $100\,000$ as well but focused on the basin of attraction). The systematic enumeration of all multiples of $6$ provides a comprehensive test.

2. Clarity: The presentation is straightforward, with clear tables and a precise description of the method.

3. Honesty: The paper correctly labels the classification as a “conjectured classification” and does not present it as a theorem. It cites the relevant theoretical works (including submitted preprints) and acknowledges that a rigorous proof is still pending.

4. Reproducibility: The attached Python scripts allow anyone to reproduce the verification.

Weaknesses

• The paper does not contain new theoretical insights; it is purely computational. However, this is acceptable for a short communication that aims to provide empirical support for a mathematical conjecture.

Overall evaluation

Computational verification plays an important role in modern number‑theoretic research, especially when a complete proof is not yet available. This paper offers strong empirical evidence that the classification is correct, and the perfect match between the exponent $m$ and the observed iteration steps adds weight to the conjecture. I therefore recommend acceptance.

Suggestions for improvement

• The authors could mention the total number of admissible numbers among all integers (not just multiples of $6$) up to $100\,000$, to give a broader picture. Since any admissible number must be a multiple of $6$, the restriction to multiples of $6$ is justified, but a brief comment would be helpful.

• A note on the computational complexity and runtime would be interesting for readers who might want to extend the verification to larger bounds.

Conclusion

The paper is a valuable contribution to the ongoing investigation of this problem. It provides robust computational confirmation of the conjectured classification and can serve as a reference for future theoretical work.