Semiperfect Number¶

A positive integer that equals the sum of a distinct subset of its proper divisors, whether or not the subset uses them all.

Version: v1 · 2026-10-03 · History
Domain-specific #: 13601
Domain group: Formal Sciences
Origin domain: Mathematics
Subdomains: Number Theory, Divisor Sums → Mathematics
Aliases: Pseudoperfect Number

Core Idea¶

A semiperfect number, also called a pseudoperfect number, is a positive integer $n$ for which some set of distinct proper divisors sums exactly to $n$. A proper divisor is a positive divisor smaller than $n$; $n$ itself may not be used, and a divisor cannot be used twice. Thus the membership rule is existential: one valid subset is enough, regardless of whether other subsets work. Benkoski and Erdős use exactly this divisor-witness definition in their original analysis of pseudoperfect and weird numbers.^[1]^[2]

The difference between total supply and exact attainability is the core abstraction. If the sum of all proper divisors is below $n$, no witness is possible. But having a sum above $n$ does not ensure that some selection equals $n$. A perfect number uses all its proper divisors to reach equality and is therefore semiperfect. A nonperfect semiperfect number omits at least one. An abundant nonmember such as 70 has enough divisor mass yet no exact selectable subset; it is called weird. The condition is not “abundant” in the strict $\sigma(n)>2n$ sense, because perfect numbers such as 6 also qualify.^[1]^[2]

This is a stable number-theoretic class, not a one-off list of integers or a general-purpose subset-sum procedure. Changing the candidate integer changes the allowed summands, because they must be divisors of that same integer. This self-referential restriction is why the form deserves a distinct identity from both Perfect Number and generic additive partitioning.

Structural Signature¶

Sig role-phrases: positive candidate integer → its proper-divisor pool → distinct subset selection → exact equality to candidate → divisor-class boundary.

Positive candidate integer. A particular $n>0$ is the object classified. The unit $1$ has no positive proper divisor and fails the witness test; the entry does not concern arbitrary real targets.^[2]
Proper-divisor pool. The allowed terms are $D(n)=\{d\in\mathbb{Z}_{>0}:d\mid n,\ d<n\}$. Admitting $n$ itself would make every positive integer trivially pass; admitting nondivisors would erase the number-theoretic restriction.
Distinct subset selection. Choose $S\subseteq D(n)$, with each $d$ used at most once. It need not be a proper subset of $D(n)$: if $S=D(n)$, the case is perfect. “Some subset” therefore includes the full set.^[1]
Exact equality. Require $\sum_{d\in S}d=n$, not merely a total at least $n$ or a close approximation. The statement $\sum_{d\in D(n)}d\ge n$ is necessary but not sufficient.
Classification boundary. The witness can coexist with full-set equality (perfect), full-set surplus (abundant nonperfect semiperfect), or fail despite surplus (weird). A further minimality condition under divisibility selects primitive semiperfect numbers.^[1]^[3]

These roles define a repeatable witness structure. A computational search may enumerate subsets or use dynamic programming, but no particular algorithm is part of the definition. Conversely, a claimed member should be accompanied by an explicit set $S$ or a proof that such a set exists.

What It Is Not¶

Semiperfect does not mean perfect. Perfect Number imposes $\sum_{d\in D(n)}d=n$; the weaker semiperfect condition needs only one selected subset. For 20, the proper divisors are $1,2,4,5,10$ and total $22$, yet $1+4+5+10=20$. Thus 20 is semiperfect and abundant, but not perfect.^[1]

It is not equivalent to being abundant. Under the usual strict convention, abundant means $\sum_{d\in D(n)}d>n$. An abundant integer can be weird; the original paper's first example is 70. Nor does every semiperfect integer have to be abundant: a perfect integer is semiperfect by using all proper divisors. Some original literature uses “abundant” inclusively at equality, so a writer must state the chosen convention before making a logical implication.^[1]

It is not identical to Primitive Semiperfect Number, already live in the encyclopedia. Primitive adds the absence of any smaller semiperfect divisor. The integer 12 has a witness but is not primitive, because 6 is semiperfect and divides 12. The integer 20 has no semiperfect proper divisor and is primitive.^[1]^[3]

Finally, it is not an unrestricted Egyptian-fraction class. Dividing a witness $n=\sum_{d\in S}d$ by $n$ yields $1=\sum_{d\in S}1/(n/d)$, but the denominators must arise as complementary divisors of one common $n$ and be distinct. That equation restates the same witness; it is not evidence of a second independent application setting.

Scope of Application¶

The definition belongs to divisor-sum number theory. It classifies positive integers by whether their available proper divisors can exactly reconstruct them using distinct terms. Its immediate neighbors are perfect, abundant, deficient, weird and primitive semiperfect numbers; each changes one logical condition about total divisor supply, exact selectable supply, or divisibility minimality. The 1974 Benkoski–Erdős paper uses the pseudoperfect/weird boundary in studying the distribution of weird numbers, rather than treating the class as merely recreational examples.^[1]

The rule can be studied computationally: an explicit subset is a positive certificate, whereas proving nonmembership may require exhausting or mathematically excluding all candidate subsets. It also has a divisibility propagation property. If $n=\sum_{d\in S}d$ and $k$ is a positive integer, then $kn=\sum_{d\in S}kd$. The $kd$ remain distinct proper divisors of $kn$, so every positive multiple of a semiperfect number is semiperfect. This elementary proof explains why the separate “primitive” subclass asks whether a member is inherited from a smaller semiperfect divisor.^[2]^[3]

No claim is made that the number class itself is used as a general optimization method, a material phenomenon or a cross-domain decision procedure. The mathematically portable piece—the existence of an exact subset witness under an eligibility constraint—could be compared with other formal domains, but would need a separately established identity.

Clarity¶

The membership question can be written compactly: does there exist $S\subseteq D(n)$ with $\sum_{d\in S}d=n$? This formulation makes three easy confusions visible. First, “proper” qualifies each divisor, not necessarily the selected subset: a perfect number may select the whole pool. Second, distinctness prevents reusing a convenient divisor until the target is reached. Third, total proper-divisor sum tells us whether enough supply exists but not whether exact equality can be achieved from the available denominations.^[1]

For 6, $D(6)=\{1,2,3\}$ and the entire pool sums to 6. For 20, $D(20)=\{1,2,4,5,10\}$ sums to 22, yet excluding 2 gives an exact witness. For 70, the proper divisors $1,2,5,7,10,14,35$ sum to 74, but there is no subset summing to 70. The three cases separate perfect inclusion, abundant inclusion, and abundant exclusion without changing the definition.^[1]^[2]

Manages Complexity¶

The witness rule reduces a broad divisor profile to a checkable proposition: is the target reachable by selecting each allowed term zero or one times? A list of divisors alone is not a conclusion; a successful selection certifies membership. This gives a clean reporting standard for numerical examples, and distinguishes a positive certificate from a mere abundance inequality.

At the same time, the rule prevents an oversimplified scalar classification. The aliquot sum $s(n)=\sum_{d\in D(n)}d$ compresses all proper divisors into one number. It immediately excludes deficient cases, where $s(n)<n$, and identifies perfect cases, where $s(n)=n$. But when $s(n)>n$, the individual divisor denominations matter. The contrast between 20 and 70 shows precisely what the scalar total discards: reachability of the target by a distinct subset.^[1]

Closure under multiplication organizes the infinite class from smaller witnesses, yet it does not imply that every member is primitive. Checking primitivity is a different question: whether a proper divisor of $n$ has the same semiperfect property. The two conditions should not be collapsed into one algorithmic shortcut.^[3]

Abstract Reasoning¶

One may express the test through binary selection variables $\epsilon_d\in\{0,1\}$ for $d\in D(n)$: $\sum_{d\in D(n)}\epsilon_d d=n$. The variables enforce distinct use. This is a constrained subset-sum equation whose candidate terms are determined by the target's divisors. It reveals why a surplus in the full sum is not decisive: the available increments can skip the required residual.

Useful deductions follow without changing the identity. If $s(n)<n$, all $\epsilon_d=1$ still fall short, hence nonmembership. If $s(n)=n$, choosing all ones proves membership and gives perfection. If $s(n)>n$, either an exact selection exists (abundant semiperfect) or not (weird). Multiplying every selected divisor by $k$ transfers a certificate from $n$ to $kn$, but the converse is not asserted: a multiple may have additional divisors that enable a witness even when the smaller number lacks one.^[1]^[2]

The reciprocal form $1=\sum_{d\in S}1/(n/d)$ follows by dividing the exact equality by $n$. It can help compare divisor identities with Egyptian-fraction notation, but cannot serve as an unlike instance: it contains precisely the same selected $S$ and a fixed relation between each denominator and $n$.

Knowledge Transfer¶

The conceptual transfer inside number theory is from aggregate threshold to exact witness. A divisor-sum condition can say that enough material exists; the existential subset equation asks whether the material can be assembled exactly under distinctness. Similar reasoning appears in constrained-combination problems, but importing the label “semiperfect” there would be a vocabulary error unless those objects really are positive integers with proper divisors. This entry transfers a diagnostic question, not the domain label.

The relationship to live Perfect Number and Primitive Semiperfect Number is asymmetric. Perfect numbers supply a special full-pool witness, and primitive semiperfect numbers impose an extra noninheritance condition; neither is a genus broad enough to parent this new identity. Once independently reviewed and promoted, a curator may consider child-to-parent relations Perfect Number → Semiperfect Number and Primitive Semiperfect Number → Semiperfect Number, with the precise DAG edge semantics checked separately.

The stripped-down pattern “eligible distinct parts exactly reconstruct a target” is an explicit future-prime question. It is not being asserted as an existing prime here: the semiperfect predicate's divisor-generated eligibility pool is what makes the named concept number-theoretic.

Examples¶

A perfect member: 6¶

The candidate integer is $n=6$. Its proper-divisor pool is $D(6)=\{1,2,3\}$. The distinct subset witness is the whole pool, $S=\{1,2,3\}$, and the exact-sum equality is $1+2+3=6$. At the classification boundary, this makes 6 both perfect and semiperfect. There is no need for a second, smaller witness set; “some subset” permits the complete pool.^[2]

Mapped back: All four constitutive roles hold, but the witness uses every available proper divisor. This tests whether the broader semiperfect definition correctly contains the live Perfect Number subclass rather than excluding it through an accidental “proper subset” reading.

An abundant but nonperfect member: 20¶

The candidate integer is $n=20$. Its proper-divisor pool is $D(20)=\{1,2,4,5,10\}$. The distinct subset witness is $S=\{1,4,5,10\}$, omitting 2. The exact-sum equality is $1+4+5+10=20$, while the whole pool totals 22. At the classification boundary, 20 is abundant and semiperfect but not perfect. It is also primitive semiperfect because its proper divisors are not semiperfect; that separate fact is not needed for its basic membership.^[1]^[3]

Mapped back: The witness is a proper subset of the eligible pool. Unlike 6, the full divisor sum overshoots; the exact selection—not the scalar sum—carries the admission. This case demonstrates the identity's additional reach beyond Perfect Number.

Near miss: 70¶

For 70 the proper-divisor pool is $\{1,2,5,7,10,14,35\}$, whose total is 74. The original paper identifies 70 as the smallest weird number: it is abundant but no distinct subset of those proper divisors totals 70. The candidate and pool roles are present, but the exact-sum witness is absent. This is a negative boundary test, not a third positive instance.^[1]

Mapped back: A surplus total does not repair a missing witness. The case prevents the class from expanding to every abundant number.

Structural Tensions¶

Total supply versus exact attainability. Summing the whole divisor pool is an efficient first screen, but discards the selectable increments; 70 shows that enough total mass need not yield exact reconstruction. Diagnostic: After the total-sum test passes, can an explicit distinct subset reach $n$ exactly?^[1]

Full-set equality versus subset existence. Requiring every proper divisor isolates Perfect Number; allowing any subset admits 20 and further numbers, while keeping the all-divisor case as a valid witness. The broader test gains coverage at the cost of losing a single scalar equality as a complete characterization. Diagnostic: Is the witness the entire pool, or does it omit eligible terms?^[1]

Inherited membership versus primitive starting points. A scaled witness proves that every multiple of a member is again semiperfect, making the class abundant with inherited instances. Primitivity cuts that propagation back to members lacking a smaller semiperfect divisor. Diagnostic: Does a proper divisor of the candidate already satisfy the subset-witness rule?^[2]^[3]

Structural–Framed Character¶

Evaluative weight: “Perfect,” “semiperfect” and “weird” sound evaluative in ordinary speech, but here each is a formal arithmetic predicate. No claim of merit, completeness of a real-world system, or oddity of a person follows.
Human-practice dependence: Mathematicians chose terminology and notation, yet whether a particular integer has the specified divisor subset is not decided by a social institution once the definition is fixed.
Institutional origin: The class is documented in original number-theory literature and curated sequences, not constituted by an administrative or legal designation.^[1]^[2]
Vocabulary travel: “Pseudoperfect” is a mathematical synonym for “semiperfect” here. The word “pseudoperfect” also appears in other named integer classes (for example, primary pseudoperfect), whose different equations must not be silently aliased.
Import versus recognition: Recognition requires the exact proper-divisor subset relation. Importing the label onto any successful subset-sum problem, or any number with a large total divisor sum, misses the entry's defining restriction.

Its character: a formal number-theoretic class, not a value judgment or institution-dependent frame. The name carries historical vocabulary; the membership relation is exact and arithmetical.

Structural Core vs. Domain Accent¶

The skeletal relation is existential exact reconstruction from eligible distinct parts. Its necessity lies in both eligibility and equality: permit arbitrary summands or approximate totals and the predicate collapses into a different problem. A general “eligible parts exactly form a target” abstraction is a future-prime question, not an actual parent-prime asserted by this package.

The domain accent is stronger than a label: eligibility is generated by the candidate integer's own positive proper divisors. The application overlay includes aliquot-sum comparisons, perfect/abundant/weird classification, multiplication closure and primitive divisibility. These are mathematical consequences and neighboring predicates; they do not turn the named number class into a domain-independent prime.^[1]^[2]

No strict live prime parent is staged. The broad ideas of selection, equality or constraint occur in the definition, but a lexical resemblance does not show that this entire typed identity is an instance of any one live prime's full signature. This workspace draft therefore records an unparented root proposal.

Within the domain-specific catalog, live Perfect number is narrower because it demands the sum of all proper divisors. Live Primitive Semiperfect Number is narrower because it demands semiperfection and no semiperfect proper divisor. After this broader identity exists canonically, those two potential child relations merit DAG review; they must not be reversed simply to avoid an unparented draft. Live deficient and related divisor-sum classes are diagnostic neighbors, not parents.

Neighborhood in Abstraction Space¶

Semiperfect Number sits in a moderately populated region (54^th percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.

Family — Integer Properties & Number-Theoretic Sets (14 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-08

Not to Be Confused With¶

Perfect number: the full proper-divisor pool sums exactly to $n$; every perfect number is semiperfect, but not conversely.^[1]
Primitive semiperfect number: a semiperfect number with no smaller semiperfect divisor; 12 is semiperfect but not primitive.^[1]^[3]
Abundant number: the complete proper-divisor sum exceeds $n$ under the strict convention; neither sufficient nor necessary for semiperfection because weird and perfect numbers occupy the two countercases.^[1]
Weird number: abundant yet not semiperfect; 70 is the standard first example.^[1]
Deficient number: its complete proper-divisor sum falls short of $n$, so no admissible subset can reach the target.
Primary pseudoperfect number or arbitrary Egyptian fraction: these have different defining equations; the unit-fraction rewriting of a semiperfect witness is a consequence, not a synonym for every such equation.

References¶

[1] S. J. Benkoski and P. Erdős, “On Weird and Pseudoperfect Numbers”, Mathematics of Computation 28, no. 126 (1974), pp. 617–623, especially abstract and definitions on p. 617. Original source for pseudoperfect, perfect, primitive pseudoperfect and weird boundaries, with the explicit $20$ witness and $70$ counterexample. registry ↩a ↩b ↩c ↩d ↩e ↩f ↩g ↩h ↩i ↩j ↩k ↩l ↩m ↩n ↩o ↩p ↩q ↩r ↩s ↩t ↩u ↩v

[2] OEIS Foundation, A005835, “Pseudoperfect (or semiperfect) numbers”, sequence title, comments and examples (accessed 30 September 2026). Curated mathematical sequence corroborating terminology, initial cases and multiplication closure. registry ↩a ↩b ↩c ↩d ↩e ↩f ↩g ↩h ↩i ↩j

[3] OEIS Foundation, A006036, “Primitive pseudoperfect numbers”, sequence definition and initial terms (accessed 30 September 2026). Corroborates the narrower divisibility-minimal subclass and the membership of 20. registry ↩a ↩b ↩c ↩d ↩e ↩f ↩g