We determine all positive integers $a_1$ for which the infinite sequence defined by $a_{n+1}=$ sum of the three largest proper divisors of $a_n$ consists entirely of numbers having at least three proper divisors. We prove that $a_1$ is admissible if and only if it can be written as $a_1 = 6\cdot12^{m}\cdot k$ with $m\ge0$, $k$ odd and $5\nmid k$. The proof closes the gap concerning divisibility by $5$ and is completely elementary.

We prove that a positive integer $a_1$ generates an infinite sequence under $a_{n+1}=$ 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 is self‑contained, uses elementary number theory, and explicitly handles the divisibility‑by‑$5$ issue that caused gaps in earlier attempts.

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 $k$.

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 $k$.

We present an extensive computational verification of the conjectured classification of admissible starting values a1 for the infinite sequence defined by a_{n+1}= sum of the three largest proper divisors of a_n. The classification asserts that a1 is admissible iff a1 = 6·12^m·k with m≥0, k odd and 5∤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.

We continue the study of the sequence defined by a_{n+1} being the sum of the three largest proper divisors of a_n. Building on the characterization of fixed points, we investigate the set of initial values a_1 that generate infinite sequences satisfying the condition. Empirical data up to 100000 suggests that all such a_1 are multiples of 6 and eventually reach a fixed point, with transients of length up to 3 observed. We present conjectures and partial results describing the basin of attraction.

We prove that any number of the form $6\cdot12^{t}k$ with $k$ odd and $5\nmid k$ generates an infinite sequence under the recurrence $a_{n+1}=$ sum of three largest proper divisors of $a_n$, each term having at least three proper divisors. This establishes the sufficiency part of the conjectured classification.

We give an alternative proof that a positive integer N with at least three proper divisors satisfies that the sum of its three largest proper divisors equals N if and only if N = 6k where k is coprime to 10. Our proof uses direct analysis of the divisor structure, complementing the reciprocal‑sum argument of [{esft}]. We also provide extensive computational verification up to 10^5.

We study the infinite sequence defined by a_{n+1} being the sum of the three largest proper divisors of a_n, where each term has at least three proper divisors. We characterize all fixed points of this iteration as numbers divisible by 6 but not by 4 or 5, and prove that any possible initial term a_1 must be divisible by 6.