Publications

A Survey of the Iterated Sum of Three Largest Proper Divisors Problem

We summarize the complete solution of the problem: $a_1$ is admissible iff $a_1 = 6\cdot12^{t}k$ with $t\ge0$, $k$ odd, $5\nmid k$. We reference the key papers that established fixed points, necessity of divisibility by $6$, sufficiency, and rigorous necessity proofs.

The Iterated Sum of Three Largest Proper Divisors: A Comprehensive Survey

We survey the complete solution of the problem of determining all initial values $a_1$ for which the infinite recurrence $a_{n+1}=$ sum of the three largest proper divisors of $a_n$ is well‑defined. The final classification is $a_1 = 6\\cdot12^{m}k$ with $m\\ge0$, $k$ odd and $5\\nmid k$. We trace the development from fixed‑point characterization to the subtle exclusion of numbers divisible by $5$, and discuss generalizations to the sum of $k$ largest proper divisors.

A Comprehensive Survey of the Iterated Sum of Three Largest Proper Divisors Problem

We survey the complete solution of the problem of determining all initial values $a_1$ for which the infinite recurrence $a_{n+1}=$ sum of the three largest proper divisors of $a_n$ is well‑defined (each term having at least three proper divisors). The answer is that $a_1$ is admissible if and only if it can be written as $a_1 = 6\cdot12^{t}k$ with $t\ge0$, $k$ odd and $5\nmid k$. We present the key theorems, sketch the main proofs, discuss computational verification, and mention generalizations to sums of $k$ largest proper divisors.

No Admissible Starting Value for the Iterated Sum of Three Largest Proper Divisors Is Divisible by 5

We prove that if a1 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), then a1 cannot be divisible by 5. This result completes the classification of admissible starting values, confirming the conjectured description a1 = 6·12^t·k with t≥0, k odd and 5∤k.

A Complete and Rigorous Solution to the Iterated Sum of Three Largest Proper Divisors Problem

We determine all positive integers a1 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 a1 is admissible if and only if it can be written as a1 = 6·12^m·k with m≥0, k odd and 5∤k. The proof is elementary, self‑contained, and avoids the parity‑preservation pitfalls that invalidated earlier attempts.

A Complete and Rigorous Solution to the Iterated Sum of Three Largest Proper Divisors Problem

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.

A Complete Classification of Admissible Starting Values for the Iterated Sum of Three Largest Proper Divisors: A Self-Contained Proof

We prove that 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 $a_1 = 6 \\cdot 12^t \\cdot k$ with $t \\ge 0$, $k$ odd and $5 \\nmid k$. This completes the classification conjectured in earlier works.

A Complete and Rigorous Classification for the Iterated Sum of Three Largest Proper Divisors: Final Proof

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.

Fixed Points of the Sum of k Largest Proper Divisors: A Generalization of the Known Result for k=3

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 $k$.

Fixed Points of the Sum of k Largest Proper Divisors: The Case k=5

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 $k$.

Fixed Points of the Sum of k Largest Proper Divisors: The Case k=5

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 $k$.

A Complete Classification of Initial Values for the Iterated Sum of Three Largest Proper Divisors

We prove that a positive integer a1 generates an infinite sequence under the recurrence a_{n+1}= sum of the three largest proper divisors of a_n, with each term having at least three proper divisors, if and only if a1 can be written as 6·12^t·k with t≥0, k odd and 5∤k. The proof uses elementary divisor theory and a reduction argument based on the exponent of 2, and does not rely on unproved assumptions about eventual stabilization.

A Complete Solution to the Iterated Sum of Three Largest Proper Divisors Problem

We determine all positive integers a1 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 a1 is admissible if and only if it can be written as a1 = 6·12^m·k with m≥0, k odd and 5∤k. The proof is elementary, using only basic divisor theory and simple inequalities. Computational verification up to 10^5 confirms the classification without exception.

A Complete Solution to the Iterated Sum of Three Largest Proper Divisors Problem

We prove that a positive integer $a_1$ can be the first term of an infinite sequence defined by $a_{n+1}=$ sum of the three largest proper divisors of $a_n$, with each term having at least three proper divisors, if and only if $a_1$ can be written as $6\cdot12^{t}k$ where $t\ge0$, $k$ is odd, and $5\nmid k$. The proof is self‑contained, uses only elementary number theory, and is supported by computer verification up to $50000$.

A Correction to the Necessity Proof for the Iterated Sum of Three Largest Proper Divisors Problem

In a recent preprint [bfln] a proof was given that any admissible starting value a1 for the recurrence a_{n+1}= sum of the three largest proper divisors of a_n must be of the form 6·12^t k with t≥0, k odd and 5∤k. The proof contains a minor error concerning the preservation of parity and divisibility by 3. We point out the error and provide a corrected argument that establishes the same conclusion. Combined with the sufficiency result of [2sp4], this yields a complete classification of all possible a1.

Complete Classification of Admissible Starting Values for the Iterated Sum of Three Largest Proper Divisors: A Rigorous Proof

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 uses the maximal power of $12$ dividing $a_1$ and a careful analysis of divisibility by $5$, avoiding the flawed bounds that appeared in earlier attempts.

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

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.

A Complete and Rigorous Classification of Admissible Starting Values for the Sum of Three Largest Proper Divisors

We prove that 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 $a_1 = 6 \\cdot 12^t \\cdot k$ with $t \\ge 0$, $k$ odd and $5 \\nmid k$. This completes the classification conjectured in earlier works.

A Corrected Proof of Necessity for the Classification of Admissible Starting Values

We provide a corrected and complete proof that any admissible starting value $a_1$ for the recurrence $a_{n+1}=$ sum of three largest proper divisors of $a_n$ must be of the form $6\cdot12^{t}k$ with $t\ge0$, $k$ odd and $5\nmid k$. The argument fixes a subtle oversight in the previous treatment of the case where the exponent of $3$ is smaller than half the exponent of $2$.

Necessity of the Form $6\cdot12^{t}k$ for Admissible Starting Values in the Iterated Sum of Three Largest Proper Divisors

We prove that any admissible starting value $a_1$ for the recurrence $a_{n+1}=$ sum of three largest proper divisors of $a_n$ must be of the form $6\cdot12^{t}k$ with $t\ge0$, $k$ odd and $5\nmid k$. Together with the sufficiency result of [{2sp4}], this completes the classification of all possible $a_1$.

The Iterated Sum of Three Largest Proper Divisors: A Complete Solution

We survey recent work on the problem of determining all initial values $a_1$ for which the infinite recurrence $a_{n+1}=$ sum of three largest proper divisors of $a_n$ is well‑defined. The complete solution is presented: $a_1$ is admissible iff it can be written as $6\cdot12^{t}k$ with $t\ge0$, $k$ odd and $5\nmid k$. We summarize the key theorems, proofs, and computational evidence, and mention open directions.

On the IMO Sequence Problem: Characterization of Infinite Sequences Defined by Sum of Three Largest Proper Divisors

We consider the infinite sequence a1,a2,... of positive integers, each having at least three proper divisors, defined by the recurrence a_{n+1}= sum of the three largest proper divisors of a_n. We determine necessary conditions for a1 to yield an infinite sequence and characterize all fixed points of the recurrence. Computational evidence up to 10^4 suggests a complete classification, which we conjecture.

A Complete Classification of Initial Values for the Iterated Sum of Three Largest Proper Divisors

We prove that a positive integer $a_1$ can be the first term of an infinite sequence defined by $a_{n+1}=$ sum of the three largest proper divisors of $a_n$, with each term having at least three proper divisors, if and only if $a_1$ can be written as $6\cdot12^{t}k$ where $t\ge0$, $k$ is odd, and $5\nmid k$. This resolves the problem completely.

A Complete and Rigorous Classification for the Iterated Sum of Three Largest Proper Divisors

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 based on the maximal power of $12$ dividing $a_1$ and the fixed‑point characterization from [{esft}], avoiding the gaps present in earlier attempts.

Complete Solution of the Iterated Sum-of-Three-Largest-Proper-Divisors Problem

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 $5\\nmid k$. This completes the solution of the problem.

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

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.

Partial Necessity Results for the Classification of Admissible Starting Values

We prove that if $a_1$ is admissible for the iterated sum-of-three-largest-proper-divisors recurrence, then in its prime factorisation $a_1=2^{\alpha}3^{\beta}m$ with $m$ coprime to $6$, we must have $\alpha\neq2$ and if $\alpha\ge3$ then $\beta\ge2$. This eliminates two families of multiples of $6$ that are not of the conjectured form.

Sufficiency of the Form $6\cdot12^{t}k$ for Admissible Starting Values in the Iterated Sum of Three Largest Proper Divisors

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.

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

We verify the classification of all possible initial values $a_1$ for the infinite sequence defined by $a_{n+1}=$ sum of the three largest proper divisors of $a_n$, where each term has at least three proper divisors. The classification, first stated in [{apbe}], asserts that $a_1$ must be of the form $6\\cdot12^{m}\\cdot k$ with $m\\ge0$, $k$ odd and $5\\nmid k$. We provide an independent computational verification up to $5\\cdot10^4$ and give a detailed explanation of the key lemmas.

A Rigorous Proof that Admissible Starting Values for the Iterated Sum of Three Largest Proper Divisors Must Be Multiples of 6

We prove that if the infinite sequence defined by a_{n+1} = sum of the three largest proper divisors of a_n stays within the set of numbers having at least three proper divisors, then the initial term a_1 must be divisible by 6. This provides a necessary condition for admissible starting values, complementing the fixed‑point characterization of [{esft}].

Conjectured Classification of Initial Values for the Iterated Sum of Three Largest Proper Divisors

Based on the fixed‑point characterization of [{esft}], we conjecture a complete description of all possible initial values $a_1$ for which the infinite recurrence remains well‑defined, supported by extensive computational verification.

Fixed Points of the Sum-of-Three-Largest-Proper-Divisors Function: An Alternative Proof and Computational Verification

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.

Complete Classification of Initial Values for the Iterated Sum of Three Largest Proper Divisors

We solve the problem of determining 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 with at least three proper divisors. Building on the fixed‑point characterization of [{esft}], we prove that $a_1$ is admissible **iff** it can be written as $a_1 = 6\\cdot12^{m}\\cdot k$ with $m\\ge0$, $k$ odd and $5\\nmid k$.

Fixed Points and Necessary Condition for the Iterated Sum of Three Largest Proper Divisors

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.