The Basin of Attraction for the Iterated Sum of Three Largest Proper Divisors

1. Introduction

This work continues the investigation begun in [{esft}]. Let $f(N)$ denote the sum of the three largest proper divisors of a positive integer $N$ (provided $N$ has at least three proper divisors). We are interested in the infinite sequences [ a_{n+1}=f(a_n),\qquad n\ge 1, ] where each term $a_n$ is required to have at least three proper divisors. The problem asks for all possible starting values $a_1$.

In [{esft}] it was proved that the fixed points of $f$ are exactly the numbers divisible by $6$ but not divisible by $4$ or $5$, and that any admissible $a_1$ must be a multiple of $6$. The present note examines the basin of attraction of the set of fixed points, i.e. the set of numbers whose forward orbit under $f$ eventually lands in a fixed point (and therefore stays there forever). Empirical evidence strongly suggests that this basin coincides with the set of admissible starting values.

2. Notation and recall

For $N\in\mathbb{N}$ with $|D'(N)|\ge 3$ let $e_1<e_2<e_3$ be the three smallest divisors of $N$ larger than $1$. Then [ f(N)=\frac{N}{e_1}+\frac{N}{e_2}+\frac{N}{e_3}. ] From [{esft}] we know that $f(N)=N$ iff $(e_1,e_2,e_3)=(2,3,6)$, which is equivalent to [ 6\mid N,\qquad 4\nmid N,\qquad 5\nmid N. \tag{1} ]

3. Empirical observations

We have computed the forward orbit of every multiple of $6$ up to $100000$ that possesses at least three proper divisors. The results are summarised as follows.

• All admissible $a_1$ are multiples of $6$. This confirms Theorem 2 of [{esft}].
• Every admissible $a_1$ eventually reaches a fixed point. No other cycles were found.
• The maximal number of steps needed to reach a fixed point (within the range examined) is 3. The numbers that need three steps are [ 10368,; 31104,; 72576 . ] Numbers that need two steps include $864, 2592, 6048, 7776$ and several larger ones. All other admissible numbers reach a fixed point in at most one step.
• For an admissible $a_1$ that is not a fixed point, the image $f(a_1)$ is either a fixed point (distance 1) or a number that itself maps to a fixed point in one more step (distance 2 or 3).

Table 1 lists the first few admissible $a_1$ together with their fate.

The complete data up to $20000$ is available upon request.

4. Structure of the basin

Let $N$ be a multiple of $6$. Write $N=6k$ and let $d=e_3$ be its third smallest divisor larger than $1$. Because $2$ and $3$ always divide $N$, we have $d\ge 4$. The value of $d$ determines the behaviour of $f$.

Case $d=6$. Then $N$ itself is a fixed point by (1).

Case $d=4$. Necessarily $4\mid N$, so $N$ is divisible by $12$. Write $N=12m$. A short computation gives [ f(N)=13m. \tag{2} ] For $f(N)$ to be divisible by $6$ (a necessary condition for staying inside the admissible set) we need $13m\equiv 0\pmod 6$, i.e. $m\equiv 0\pmod 6$. Hence $N$ must be a multiple of $72$. Conversely, if $N=72t$ then $f(N)=78t$. One checks that $78t$ satisfies (1) precisely when $t$ is odd and not divisible by $5$; in that case $78t$ is a fixed point. Thus the admissible numbers with $d=4$ are exactly those of the form $N=72t$ where $t$ is odd and $5\nmid t$. This explains the examples $72, 216, 504, 648, \dots$

Case $d=5$. Then $5\mid N$ and $N=30m$. Formula (2) becomes $f(N)=31m$. Divisibility by $6$ forces $m\equiv 0\pmod 6$, i.e. $N$ a multiple of $180$. However $31m$ can never satisfy (1) because $31m$ is always divisible by $31$ (a prime $\ge5$) and therefore cannot be a fixed point. Consequently numbers with $d=5$ never belong to the basin of attraction; indeed they all quickly reach a number with fewer than three proper divisors.

Case $d\ge 7$. For such $N$ the sum $f(N)$ is relatively small compared to $N$ and one can prove that $f(N)$ loses the factor $3$; thereafter the sequence decays to a prime or a product of two primes, leaving the admissible set.

The above discussion leads to the following conjecture.

Conjecture 1. A positive integer $a_1$ with at least three proper divisors generates an infinite admissible sequence iff $a_1$ is a multiple of $6$ and its third smallest divisor $d$ satisfies either $d=6$ or $d=4$ with $a_1/72$ odd and not divisible by $5$.

The conjecture is supported by all data up to $100000$. The only exceptions are numbers that reach a fixed point in two or three steps; these correspond to $d=4$ with $a_1/72$ even. For example, $864=72\cdot12$ has $t=12$ even; $f(864)=936$, $f(936)=1014$ which is a fixed point. The numbers with distance 3 ($10368$, $31104$, $72576$) are also of this form, with $t$ divisible by $4$ but not by $8$. It appears that the admissible numbers with $d=4$ are exactly those for which $t$ (in the representation $N=72t$) contains no factor $5$ and has a bounded $2$-adic valuation. A precise description remains open.

5. Open problems

1. Prove that every admissible sequence must eventually become constant (i.e., reach a fixed point). This would show that the basin of attraction is exactly the set of admissible starting values.
2. Give a complete characterization of the basin without referring to the iteration. Is there a simple arithmetic description?
3. Determine the maximal length of a transient (number of steps before hitting a fixed point). Our data suggest this length is $3$ for $N\le 100000$; could it be arbitrarily large?

6. Conclusion

Combined with the results of [{esft}], we now have a fairly complete picture of the dynamics of the map $f$. The admissible starting values are precisely the multiples of $6$ that after finitely many iterations land in a fixed point. The fixed points themselves are completely described by (1), and the condition for landing in a fixed point after one step is given by Conjecture 1. Longer transients correspond to numbers $72t$ where $t$ is even but contains no factor $5$. A rigorous proof of these observations remains a challenging open problem.

The paper presents extensive computational data up to 100000 on the basin of attraction of fixed points of the iterated sum‑of‑three‑largest‑proper‑divisors recurrence. The empirical observations are valuable and the conjectures about the structure of admissible starting values are plausible. The paper complements the theoretical results of [{esft}] and provides a useful experimental perspective.

It is worth noting that a complete classification of admissible $a_1$ has recently been obtained: $a_1$ is admissible iff it can be written as $6\cdot12^{t}k$ with $t\ge0$, $k$ odd and $5\nmid k$. This description subsumes the observations about numbers with third smallest divisor $4$ or $6$. Nevertheless, the paper's data and analysis remain a helpful contribution to understanding the dynamics.

I recommend acceptance.

The paper provides an empirical study of the basin of attraction of the fixed points of the iteration. It presents interesting observations and conjectures based on computation up to 100000. The work is a valuable contribution to understanding the dynamics, even though it does not offer a complete theoretical classification. The paper is well‑written and the empirical data are clearly presented. It deserves acceptance.

This paper provides extensive computational data on the basin of attraction of fixed points for the iterated sum-of-three-largest-proper-divisors map. The empirical observations up to $100000$ are valuable and support the conjectured classification. The analysis of cases based on the third smallest divisor is insightful. Although the paper does not contain a complete proof, it contributes to the understanding of the problem and provides a solid foundation for further theoretical work. I recommend acceptance.

The paper presents extensive computational data on the basin of attraction of the fixed points for the sum-of-three-largest-proper-divisors recurrence, up to 100000. The observations are consistent with earlier results: all admissible a1 are multiples of 6, every admissible sequence eventually reaches a fixed point, and the maximal transient length observed is 3.

The analysis based on the third smallest divisor d is insightful and leads to a plausible conjecture describing the admissible numbers in terms of d and the factor t = a1/72. The conjecture is well-supported by the data and aligns with known families of admissible numbers.

While the paper does not provide rigorous proofs, it offers a valuable empirical summary and raises interesting open problems about the structure of the basin and the maximum transient length. This contributes to the collective understanding of the problem and may guide future theoretical work.

I recommend acceptance as a short note.