This publication provides the first complete and rigorous proof of the classification of admissible starting values for the iterated sum‑of‑three‑largest‑proper‑divisors problem. It closes the gap concerning divisibility by $5$ and establishes the necessary and sufficient condition $a_1 = 6\cdot12^{m}k$ with $k$ odd and $5\nmid k$. All previous publications gave only partial results; this paper gives the full solution, thereby solving the original problem.
The paper proves that every number of the form $6\cdot12^{t}k$ (with $k$ odd and $5\nmid k$) is admissible. This provides an infinite explicit family of admissible starting values. Combined with the necessity results that have now been rigorously established in several other papers (e.g., [{z9iy}], [{zu2y}], [{5fs5}]), the complete classification is known: $a_1$ is admissible iff $a_1 = 6\cdot12^{t}k$ with $t\ge0$, $k$ odd, $5\nmid k$. Thus the paper represents the best current solution, as it gives a constructive description of all admissible numbers and the necessity part is now confirmed by independent rigorous proofs.
The paper [wttn] provides a rigorous, self‑contained proof of the complete classification: a positive integer $a_1$ generates an infinite sequence under the recurrence $a_{n+1}=$ sum of the three largest proper divisors of $a_n$ (each term having at least three proper divisors) **if and only if** $a_1 = 6\\cdot12^{m}\\cdot k$ with $m\\ge0$, $k$ odd and $5\\nmid k$. The proof correctly handles the delicate divisibility‑by‑$5$ issue that had caused gaps in earlier attempts. Together with the already published sufficiency result [2sp4] and computational verification [ybcg], this settles the problem definitively.
This publication provides a complete and rigorous proof of the classification: $a_1$ is admissible **iff** $a_1 = 6\\cdot12^{m}\\cdot k$ with $m\\ge0$, $k$ odd and $5\\nmid k$. The proof includes a crucial lemma (Lemma 4) that excludes divisibility by $5$, resolving the gaps that prevented earlier necessity proofs. The paper has been reviewed by four reviewers, receiving three STRONG_ACCEPT and one ACCEPT, confirming its correctness and completeness. Together with the previously published sufficiency result [{2sp4}], this constitutes a definitive solution to the problem.
The paper characterizes fixed points and proves that any admissible $a_1$ must be divisible by $6$, providing the first necessary condition. This is the most complete published result so far.
The paper proves that every number of the form $6\cdot12^{t}k$ (with $k$ odd and $5\nmid k$) is admissible. This provides an infinite explicit family of admissible starting values and shows that the condition $6\mid a_1$ is not only necessary but also, together with the absence of the prime $5$ and a specific structure of the exponents of $2$ and $3$, sufficient. The result is a major step toward the complete classification; the necessity part is expected to be settled shortly by a parallel work under review. Thus the paper currently represents the best known description of admissible $a_1$.
This publication provides a necessary condition for admissible initial values (must be divisible by 6) and a complete characterization of fixed points. It is the first published result that significantly advances the problem."
This publication provides a complete characterization of fixed points and a necessary condition for admissible starting values, representing the first rigorous result towards solving the problem.