#
Equisum Partitions of Sets of Positive Integers^{ †}

^{†}

## Abstract

**:**

## 1. Introduction

- If $b,n\in {\mathbb{Z}}^{+},2\le b\le \lceil n/2\rceil $ and $b\le 12$ or $b$ is any prime-power ${p}^{a},$ the initial interval of integers $\left[1,n\right]$ has an equisum $b$–partition if and only if $\mathsf{\Sigma}\left[1,n\right]$ is a multiple of $b$.
- No product of two odd prime-powers has an equisum $2$–partition of its positive divisors; however, for any prime $p\ge 3$ and $a,m\in {\mathbb{Z}}^{+}$ with $m$ odd, the set of all positive divisors of ${2}^{a}{p}^{m}$ has an equisum $2$–partition when ${2}^{a+1}\ge \sigma \left({p}^{m}\right)$. Even perfect numbers are the “boundary case” of this result.
- If the set of aliquot divisors of $n\in {\mathbb{Z}}^{+}$ has an equisum $2$–partition then $n$ has at least two distinct prime factors. For any prime $p\ge 3$ and $a,m\in {\mathbb{Z}}^{+}$ with $m$ odd, the set of aliquot divisors of ${2}^{a}{p}^{m}$ has an equisum $2$–partition if ${2}^{a+1}\ge \sigma \left({p}^{m}\right)$. Again, even perfect numbers are the “boundary case” of this result.
- If $n\in {\mathbb{Z}}^{+}$ is odd, its set of aliquot divisors can have an equisum $2$–partition only when $n$ is a perfect square. Further, if $n$ has exactly two distinct prime factors $p,q,$ they must either be twin primes or 3 and 7. The aliquot divisors always have an equisum $2$–partition when $\left\{p,q\right\}=\left\{3,5\right\}$; this probably also holds when $\left\{p,q\right\}=\left\{3,7\right\}$. However, there may be only finitely many pairs $\left\{p,q\right\}$ such that the aliquot divisors of $n$ have such a partition.

## 2. Initial Intervals of ${\mathbb{Z}}^{+}$

**Example**

**1.**

**Example**

**2.**

**Direct Sum Construction:**

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Lemma**

**1.**

**Proof.**

**Theorem**

**3.**

**Lemma**

**2.**

**Proof.**

**Lemma**

**3.**

**Theorem**

**4.**

**Theorem**

**5.**

**Corollary**

**1.**

**Example**

**3.**

**Lemma**

**4.**

**Proof.**

**Example**

**4.**

**Example**

**5.**

**Theorem**

**6.**

## 3. Initial Intervals of $\mathbb{P}$

**Example**

**6.**

**Conjecture:**

## 4. Divisor Sets

**Lemma**

**5.**

**Proof.**

**Theorem**

**7.**

## 5. Aliquot Sets

**Theorem**

**8.**

**Proof.**

**Theorem**

**9.**

**Proof.**

**Theorem**

**10.**

**Proof.**

**Theorem**

**11.**

**Proof.**

## 6. Concluding Remarks

## Footnote

## Funding

## Conflicts of Interest

## References

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**MDPI and ACS Style**

Eggleton, R.B. Equisum Partitions of Sets of Positive Integers. *Algorithms* **2019**, *12*, 164.
https://doi.org/10.3390/a12080164

**AMA Style**

Eggleton RB. Equisum Partitions of Sets of Positive Integers. *Algorithms*. 2019; 12(8):164.
https://doi.org/10.3390/a12080164

**Chicago/Turabian Style**

Eggleton, Roger B. 2019. "Equisum Partitions of Sets of Positive Integers" *Algorithms* 12, no. 8: 164.
https://doi.org/10.3390/a12080164