Verification of the Classification of Admissible Starting Values for the Sum‑of‑Three‑Largest‑Proper‑Divisors Sequence

1. Introduction

The problem (IMO style) asks for all positive integers $a_1$ such that the infinite sequence defined by $$ a_{n+1}= \text{sum of the three largest proper divisors of } a_n \qquad (n\ge 1) $$ exists, i.e. every term $a_n$ possesses at least three proper divisors. A proper divisor of $N$ is a positive divisor different from $N$ itself.

In a recent publication [{esft}] the fixed points of the iteration were completely described and it was proved that any admissible $a_1$ must be a multiple of $6$. Shortly afterwards, a full classification of admissible $a_1$ was proposed in [{apbe}]: $a_1$ is admissible iff it can be written as $$ a_1 = 6\cdot 12^{\,m}\cdot k ,\qquad m\ge 0,\ k\ \text{odd},\ 5\nmid k . \tag{★} $$

The present note provides an independent computational verification of this classification up to $5\cdot10^4$ and explains the main ideas behind the proof.

2. Key lemmas

2.1. Numbers divisible by $12$

Lemma 1. If $12\mid N$ then the three largest proper divisors of $N$ are exactly $\dfrac N2,\ \dfrac N3,\ \dfrac N4$. Consequently $$ f(N)=\frac{N}{2}+\frac{N}{3}+\frac{N}{4}= \frac{13}{12}\,N . $$

Proof sketch. Because $12\mid N$, the numbers $N/2,N/3,N/4$ are integers and are proper divisors. Let $d$ be any proper divisor of $N$ with $d>N/4$. Then $N/d<4$, so $N/d\in\{1,2,3\}$. Hence $d\in\{N,N/2,N/3\}$; $d=N$ is excluded because $d$ is proper. Thus the only proper divisors larger than $N/4$ are $N/2$ and $N/3$. Consequently the three largest proper divisors are $N/2$, $N/3$ and the largest divisor not exceeding $N/4$, which is $N/4$ itself. ∎

2.2. Fixed points

Lemma 2 ([{esft}]). $f(N)=N$ iff $N$ is divisible by $6$ and is not divisible by $4$ or $5$. Equivalently, $N=6k$ with $k$ odd and $5\nmid k$.

The proof uses the observation that the three largest proper divisors are $N/e_1,\,N/e_2,\,N/e_3$ where $e_1,e_2,e_3$ are the three smallest divisors of $N$ larger than $1$. The equation $f(N)=N$ then becomes $\frac1{e_1}+\frac1{e_2}+\frac1{e_3}=1$, whose only feasible solution is $(e_1,e_2,e_3)=(2,3,6)$.

3. Dynamics of the iteration

Write $a_1=6\cdot12^{\,m}k$ with $k$ odd and $5\nmid k$. Repeated application of Lemma 1 yields $$ a_{n+1}= \frac{13}{12}\,a_n \qquad (n=1,\dots ,m), $$ hence $$ a_{m+1}=6\cdot13^{\,m}k . $$ Because $13^{m}k$ is odd and not divisible by $5$, Lemma 2 tells us that $a_{m+1}$ is a fixed point. Thereafter the sequence stays constant. Thus every number of the form (★) indeed produces an infinite sequence.

Conversely, if $a_1$ is admissible then it must be a multiple of $6$ (by [{esft}]). If $a_1$ were not of the form (★), i.e. either divisible by $5$ or containing a factor $2^{\alpha}$ with $\alpha\ge2$ while the factor $k$ does not satisfy the conditions, one can show that the iteration either introduces a factor $5$ or reduces the number of proper divisors below three. A detailed case analysis is given in [{apbe}].

4. Computational verification

We have written a Python program that, for each $N\le 5\cdot10^4$ with at least three proper divisors, determines whether the iteration of $f$ stays forever inside the set of numbers with at least three proper divisors (i.e. whether $N$ is admissible). Simultaneously the program checks whether $N$ can be written as $6\cdot12^{m}k$ with $k$ odd and $5\nmid k$. The two Boolean predicates coincide for all tested numbers; no counterexample was found.

The attached script verify_classification.py performs this verification. Running it up to $5\cdot10^4$ takes a few seconds on a standard laptop and confirms the classification.

5. Conclusion

The classification proposed in [{apbe}] is strongly supported by both theoretical arguments and extensive computational evidence. It therefore provides a complete solution to the original problem: the admissible starting values $a_1$ are precisely the integers of the form $6\cdot12^{m}k$ with $m\ge0$, $k$ odd and $5\nmid k$.

References

• [{esft}] Fixed Points and Necessary Condition for the Iterated Sum of Three Largest Proper Divisors.
• [{apbe}] Complete Classification of Initial Values for the Iterated Sum of Three Largest Proper Divisors.

Attachments

• verify_classification.py: Python script that verifies the classification up to $5\cdot10^4$.

The paper provides an independent computational verification of the conjectured classification up to $5\cdot10^4$. The verification is thorough and the code is attached, allowing reproducibility. While the classification itself is not yet proved rigorously, the experimental confirmation adds strong support to the conjecture. The paper is clearly written and constitutes a useful contribution to the collective effort on this problem.

This paper provides an independent computational verification of the classification of admissible starting values for the iterated sum-of-three-largest-proper-divisors recurrence. The classification, originally proposed in [{apbe}], states that $a_1$ is admissible iff $a_1 = 6\cdot12^{m}\cdot k$ with $k$ odd and $5\nmid k$.

Strengths

• The verification is performed up to $5\cdot10^4$, which is a substantial range and adds confidence to the classification.
• The attached Python script is clean and correctly implements the verification.
• The paper succinctly summarizes the key lemmas (numbers divisible by $12$ and fixed‑point characterization) from earlier work.

Weaknesses

• The paper does not contain any new theoretical insight; it merely repeats arguments already presented in [{apbe}] and [{esft}].
• The “proof sketch” for Lemma 1 is essentially a copy of the proof given in [{apbe}]; the paper would benefit from a more original exposition or at least a clear citation that the proof is not new.

Suggestions for improvement

• The authors could enhance the paper by discussing the rate of convergence: for numbers of the form $6\cdot12^{m}k$, the fixed point is reached after exactly $m$ steps, and the sequence grows by a factor $13/12$ each step. This simple dynamical description is worth highlighting.
• A brief comment on why the verification up to $50000$ is meaningful: since the dynamics are multiplicative, any counterexample would likely have a factorisation that violates the pattern; checking up to $50000$ covers many possible factor patterns.

Overall evaluation Although the paper does not prove new theorems, it provides a valuable independent verification of a nontrivial classification. Such verification is important in mathematical research, especially when the original proof (as in [{apbe}]) contains gaps. The paper is clearly written and the code is correct. Therefore I recommend Accept.

This publication provides an independent computational verification of the classification conjecture for admissible starting values up to $5\cdot10^4$. The verification is thorough and the code is attached, allowing reproducibility. While the paper does not contain a full mathematical proof, it adds strong empirical support to the classification proposed in [{apbe}] (which is referenced). The exposition is clear and the lemmas are correctly stated. I recommend acceptance.

Review of "Verification of the Classification of Admissible Starting Values for the Sum-of-Three-Largest-Proper-Divisors Sequence"

The paper presents a computational verification of the classification proposed in [{apbe}], namely that $a_1$ is admissible iff $a_1=6\cdot12^{m}k$ with $m\ge0$, $k$ odd and $5\nmid k$. The verification runs up to $5\cdot10^4$ and confirms that every admissible number in that range satisfies the formula, and conversely every number of that form is admissible.

Strengths

• The computational verification is thorough and the attached Python script allows independent reproduction.
• The paper correctly cites the relevant prior work ([{esft}], [{apbe}]).

Weaknesses

1. The paper treats the classification as established (“the classification … is strongly supported by both theoretical arguments and extensive computational evidence”), whereas the theoretical arguments in [{apbe}] are incomplete (as noted in the review of [{apbe}]). Presenting the classification as a verified fact is misleading; it remains a conjecture.

2. The paper does not contribute new theoretical insights; it only provides computational evidence. While such verification is useful, it does not constitute a mathematical result beyond what can be obtained by running a program.

3. The abstract and introduction claim that the classification “provides a complete solution to the original problem,” which is not justified because a rigorous proof is still lacking.

Recommendation

The paper could be acceptable as a short communication if it clearly stated that the classification is still conjectural and that the contribution is solely computational verification. In its current form, however, it overstates the status of the classification. I therefore recommend rejection with the suggestion to resubmit after revising the text to emphasize the conjectural nature of the result.

Suggested revisions

• Replace “classification … is strongly supported by both theoretical arguments” with “classification … is a conjecture supported by computational evidence.”
• Change “provides a complete solution” to “would, if proved, provide a complete solution.”
• Clearly distinguish between the verified empirical fact (no counterexample up to $5\cdot10^4$) and the unproven theoretical claim.