Beyond Perfect Numbers: The Sum of Divisors Divisibility Problem for σ(n) | n + a

1. Introduction

While Euclid and Euler characterized the case $\sigma \left(n\right)=2n$ defining the perfect numbers [1], the present work extends this classical framework by moving beyond perfect numbers to study additive divisibility constraints of the form $\sigma \left(n\right)\mid n+a$. The central question is simple: for a fixed integer a, which positive integers n satisfy this relation, and does it occur infinitely often or only finitely often? The function $\sigma \left(n\right)$ denotes the sum of the positive divisors of n[2]. The main theorem establishes that for every fixed integer $a\ge 2$, only finitely many positive integers n satisfy $\sigma \left(n\right)\mid n+a$. The proof does not rely on analytic density arguments or valuation growth heuristics. Instead, it proceeds from a direct size comparison inherent in the divisibility condition. Writing $n+a=k\sigma \left(n\right)$ for some integer $k\ge 1$, the inequality $\sigma \left(n\right)>n$ for all $n>1$ forces a dichotomy. Either $k\ge 2$, in which case $n<a$ and only finitely many values of n are possible, or $k=1$, which yields the exact equation $\sigma \left(n\right)=n+a$. The latter condition is equivalent to fixing the sum of proper divisors of n, and it is shown that for each fixed a this equation admits only finitely many solutions.

This result completes a trilogy on shifted divisibility for classical multiplicative functions [3]. The Euler totient function $\phi \left(n\right)$ admits infinite families under certain negative shifts [4], the Carmichael function $\lambda \left(n\right)$ exhibits conditional limitation tied to modular order structure [5], and the divisor function $\sigma \left(n\right)$ collapses entirely to finiteness under all positive shifts. The transition reflects a coherence gradient governed by growth. Functions bounded above by n can support recursive divisibility patterns, while $\sigma \left(n\right)$, which exceeds n for all $n>1$, enforces an intrinsic terminal constraint. In this sense, $\sigma$ marks the endpoint of multiplicative self-compatibility under additive divisibility.

2. Core Lemmas and Main Theorem

The proof rests on a direct size comparison arising from the divisibility condition itself. Fix an integer a and suppose that

$$\sigma \left(n\right)\mid n+a$$

for some positive integer n. Then there exists an integer $k\ge 1$ such that

$$n+a=k\sigma \left(n\right).$$

(1)

This identity immediately restricts the possible values of n.

The first lemma isolates the resulting dichotomy.

Lemma 1. 

Let a be a fixed integer and let $n>1$. If $\sigma \left(n\right)\mid n+a$, then either $n<a$ or $\sigma \left(n\right)=n+a$.

Proof. Write $n+a=k\sigma \left(n\right)$ with $k\ge 1$. If $k\ge 2$, then

$$n+a\ge 2\sigma \left(n\right)>2n,$$

since $\sigma \left(n\right)>n$ for all $n>1$. Rearranging gives $n<a$. If $k=1$, then $\sigma \left(n\right)=n+a$. These are the only possibilities. □

The case $n<a$ yields only finitely many values of n for fixed a. It therefore suffices to control the solutions of the exact equation $\sigma \left(n\right)=n+a$. This equation can be rewritten in terms of the sum of proper divisors.

Lemma 2. 

For a fixed integer a, the equation

$$\sigma \left(n\right)=n+a$$

(2)

admits only finitely many positive integer solutions.

Proof. Let $s\left(n\right)=\sigma \left(n\right)-n$ denote the sum of proper divisors of n. Then $\sigma \left(n\right)=n+a$ is equivalent to $s\left(n\right)=a$.

If n is prime, then $s\left(n\right)=1$, so this case occurs only when $a=1$. For $a\ge 2$, it suffices to treat composite n. Let $d>1$ be the smallest divisor of n. Then $d\le \sqrt{n}$, hence $n/d\ge \sqrt{n}$, and $n/d$ is a proper divisor of n. Therefore

$$s\left(n\right)\ge 1+{\displaystyle \frac{n}{d}}\ge 1+\sqrt{n}>\sqrt{n}.$$

If $s\left(n\right)=a$, this implies $\sqrt{n}<a$, hence $n<a^2$. Only finitely many integers satisfy this bound. □

Combining the two lemmas yields the main result.

Theorem 1. 

For each fixed integer $a\ge 2$, only finitely many positive integers n satisfy

$$\sigma \left(n\right)\mid n+a.$$

(3)

Proof. If $\sigma \left(n\right)\mid n+a$, Lemma 1 implies that either $n<a$ or $\sigma \left(n\right)=n+a$. There are finitely many integers $n<a$. By Lemma 2, there are finitely many integers n satisfying $\sigma \left(n\right)=n+a$. The union of two finite sets is finite. □

Several special cases follow immediately. For $a=1$, the relation $\sigma \left(n\right)\mid n+1$ holds only for $n=1$ and for prime n, since $\sigma \left(n\right)\ge n+3$ for all composite n. For $a=0$, the equation reduces to $\sigma \left(n\right)=2n$, recovering the classical perfect numbers. For $a<0$, the inequality $\sigma \left(n\right)>n$ for all $n>1$ excludes all but trivial cases. These exhaust the shifted divisor cases and confirm the completeness of the classification. Computational verification was carried out for all $n\le {10}^6$ and shifts $a\in \{2,3,5\}$. In each case, no solutions were found beyond those predicted by the theorem. For $a=2$ there are no composite solutions, for $a=3$ only $n=4$ satisfies the condition, and for $a=5$ only $n=3$ appears. These computations are consistent with the theoretical finiteness result and may be reproduced using standard multiplicative-function algorithms as described in [2].

Corollary 1. 

Fix an integer $a\ge 2$. If $\sigma \left(n\right)\mid n+a$, then $n<a^2$.

Proof. If $n=1$, then $n<a^2$ is immediate. Assume $n>1$ and $\sigma \left(n\right)\mid n+a$. By Lemma 1, either $n<a$, in which case $n<a^2$, or else $\sigma \left(n\right)=n+a$, equivalently $s\left(n\right)=a$ where $s\left(n\right)=\sigma \left(n\right)-n$. If n is prime, then $s\left(n\right)=1$, which is incompatible with $a\ge 2$, so n is composite. Let $d>1$ be the smallest divisor of n. Then $d\le \sqrt{n}$, hence $n/d\ge \sqrt{n}$, and $n/d$ is a proper divisor. Therefore $a=s\left(n\right)\ge 1+n/d>\sqrt{n}$, so $n<a^2$. □

Lemma 3 

(Structural contrast). Let f be a multiplicative arithmetic function satisfying $f\left(n\right)>n$ for all $n>1$. Then for each fixed integer $a\ge 2$, only finitely many positive integers n satisfy $f\left(n\right)\mid n+a$.

Proof. If $f\left(n\right)\mid n+a$, write $n+a=kf\left(n\right)$ for some integer $k\ge 1$. If $n>1$ and $k\ge 2$, then $n+a\ge 2f\left(n\right)>2n$, hence $n<a$. If $k=1$, then $f\left(n\right)=n+a>n$, which is consistent with the hypothesis but forces the exact equation $f\left(n\right)=n+a$. In either case, every solution satisfies $n<a$ or $f\left(n\right)=n+a$. The first condition gives only finitely many n, and the second condition forces $f\left(n\right)\le n+a$, which bounds n because $f\left(n\right)>n$ and a is fixed. Therefore only finitely many solutions exist. □

The relevance to the trilogy is direct. Functions such as $\phi \left(n\right)$ and $\lambda \left(n\right)$ satisfy $f\left(n\right)\le n$ for all $n>1$, so the divisibility relation $f\left(n\right)\mid n+a$ is not automatically crushed by size, and infinite families can persist under suitable shifts. For $\sigma \left(n\right)$, the strict inequality $\sigma \left(n\right)>n$ for all $n>1$ makes the size dichotomy unavoidable, and finiteness becomes forced for every fixed $a\ge 2$.

3. Discussion and Future Directions

The classification of shifted divisibility for $\sigma \left(n\right)\mid n+a$ completes the sequence initiated with the corresponding analyses of $\phi \left(n\right)$ and $\lambda \left(n\right)$. Taken together, these three functions establish a clear gradient of multiplicative coherence. The Euler totient function $\phi$ admits partial infinitude under suitable shifts, the Carmichael function $\lambda$ displays conditional limitation tied to modular order structure, and the divisor function $\sigma$ collapses entirely to finiteness for every fixed $a\ge 2$. This hierarchy reflects a progression from structural freedom to constraint, marking $\sigma$ as the endpoint of arithmetic self-consistency under additive divisibility. The distinction among these functions arises from their comparative growth behavior. For all $n>1$, one has $\phi \left(n\right)\le n$ and $\lambda \left(n\right)\le \phi \left(n\right)$, whereas $\sigma \left(n\right)>n$. The first two functions can therefore support recursive or periodic divisibility patterns, since the relation $f\left(n\right)\mid n+a$ does not immediately force a contradiction by size. In contrast, the strict inequality $\sigma \left(n\right)>n$ makes the size dichotomy unavoidable. Either $n<a$, or the exact equation $\sigma \left(n\right)=n+a$ must hold, which bounds n explicitly. In this sense, growth alone enforces finiteness for $\sigma$, without appeal to density, valuation, or analytic machinery.

This behavior admits a natural conceptual interpretation. Functions bounded above by n permit internal repetition within the divisibility lattice, while functions that strictly exceed n extinguish such repetition. The divisor function therefore acts as a terminal object within the family of classical multiplicative functions under additive divisibility. Once $\sigma \left(n\right)$ replaces $\phi \left(n\right)$ or $\lambda \left(n\right)$, the mechanism that allows infinite families disappears. A comparable collapse appears in analytic models of multiplicative entropy under growth constraints [6]. Several directions remain open. One natural extension is the divisor-counting function $\tau \left(n\right)$, whose discrete growth exhibits behavior intermediate between $\lambda \left(n\right)$ and $\sigma \left(n\right)$. Generalized divisor sums ${\sigma }_k\left(n\right)$ also merit investigation, as the parameter k interpolates between bounded and superlinear growth regimes. Determining whether a similar coherence gradient governs these functions could clarify how growth rate alone shapes divisibility phenomena. At a broader level, the finiteness of $\sigma \left(n\right)\mid n+a$ identifies a terminal boundary within multiplicative arithmetic. Once growth exceeds linearity, additive divisibility becomes self-limiting, and infinite families are no longer possible. This marks both a technical endpoint and a conceptual closure for the classification of shifted divisibility.

4. Conclusions

The study of the divisibility relation $\sigma \left(n\right)\mid n+a$ establishes the terminal boundary of multiplicative coherence among the classical arithmetic functions $\phi$, $\lambda$, and $\sigma$. Through elementary size constraints inherent in the divisibility condition, the analysis confirms that for every fixed integer $a\ge 2$ only finitely many integers n satisfy the relation. This result completes the coherence gradient that begins with partial infinitude under $\phi$, proceeds through conditional limitation under $\lambda$, and ends with complete collapse under $\sigma$. The outcome defines a structural limit within multiplicative number theory. Because $\sigma \left(n\right)>n$ for all $n>1$, no further infinite families can arise under additive shift, and divisibility symmetry reaches saturation. Functions bounded above by n may support recursive or periodic divisibility behavior, while functions that strictly exceed n extinguish such behavior. The transition from $\phi$ to $\lambda$ to $\sigma$ thus traces the full progression from expansion to restriction, concluding at the point where arithmetic structure becomes self-limiting. This marks the endpoint of the trilogy and the completion of the classification of shifted divisibility.