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.
Core Idea¶
A semiperfect number, or pseudoperfect number, is a positive integer \(n\) that equals the sum of a set of distinct proper divisors of \(n\). A proper divisor is smaller than \(n\), but the selected set may be the full collection of proper divisors. Thus every perfect number is semiperfect, while numbers such as 20 show that the reverse is false: \(20=1+4+5+10\) even though its complete proper-divisor sum is 22.[ref-f65483581b88][ref-09ef9967d673]
The abstraction is an exact subset witness. Knowing that the proper divisors add to at least \(n\) is only a necessary supply check, not an exact membership test. The abundant number 70 has a total proper-divisor sum of 74 but no subset summing to 70, so it is weird rather than semiperfect. Strict abundance is not required, because perfect numbers such as 6 pass.[^ref-f65483581b88]
Scope of Application¶
This is a number-theoretic class for positive integers, not a general subset-sum algorithm. Its admissible terms are generated by the candidate number's own divisor relation. It helps distinguish perfect, abundant semiperfect, weird and deficient integers. A narrower live class, Primitive Semiperfect Number, additionally requires that no proper divisor be semiperfect; 12 is a nonprimitive member because 6 divides it.[ref-f65483581b88][ref-22c0b5780bb5]
Clarity¶
Write \(D(n)=\{d>0:d\mid n,\ d<n\}\). The question is whether some \(S\subseteq D(n)\) satisfies \(\sum_{d\in S}d=n\). For 6, \(S=D(6)=\{1,2,3\}\); for 20, \(S=\{1,4,5,10\}\) omits 2 from \(D(20)\). These are unlike subclasses of the same arithmetic relation: perfect/full-set and abundant/proper-subset. Dividing a witness by \(n\) gives a distinct-unit-fraction identity, but merely rewrites this same test rather than supplying a second setting.[ref-f65483581b88][ref-09ef9967d673]
Manages Complexity¶
An explicit witness certifies membership without requiring a special search algorithm. The full proper-divisor sum gives a fast screen: below \(n\) excludes membership and equality proves perfection. Above \(n\), individual divisor denominations still matter; the 20/70 contrast shows why a scalar total cannot decide exact attainability. Multiplying a witness by a positive integer produces a witness for the corresponding multiple, explaining how nonprimitive members arise.[ref-f65483581b88][ref-09ef9967d673]
Abstract Reasoning¶
The formal test is \(\sum_{d\in D(n)}\epsilon_d d=n\) with each \(\epsilon_d\in\{0,1\}\). Binary coefficients enforce distinctness. If every coefficient can be one and the sum is \(n\), the number is perfect; if some selection works after the full sum exceeds \(n\), it is nonperfect semiperfect; if none works despite surplus, it is weird. Neither abundance nor multiplication closure replaces the subset-witness requirement.[^ref-f65483581b88]
Knowledge Transfer¶
The portable diagnostic is to separate aggregate supply from exact assembly under eligibility constraints. The name “semiperfect” itself should not be imported outside positive-integer divisor classification. A broader exact-subset-reconstruction skeleton is a future-prime question, not a declared existing prime. This draft is staged without a live DAG parent: Perfect Number and Primitive Semiperfect Number are potentially narrower children to assess after canonical promotion, not parents of the broader identity.
[^ref-f65483581b88]: S. J. Benkoski and P. Erdős, “On Weird and Pseudoperfect Numbers”, Mathematics of Computation 28 (1974), p. 617 definitions and examples. [^ref-09ef9967d673]: OEIS Foundation, A005835, “Pseudoperfect (or semiperfect) numbers”, definition, comments and examples (accessed 30 September 2026). [^ref-22c0b5780bb5]: OEIS Foundation, A006036, “Primitive pseudoperfect numbers”, definition and initial terms (accessed 30 September 2026).
Neighborhood in Abstraction Space¶
Semiperfect Number sits in a moderately populated region (54th percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.
Family — Integer Properties & Number-Theoretic Sets (14 abstractions)
Nearest neighbors
- Trial Division — 0.89
- Primitive Semiperfect Number — 0.88
- Square-Free Integer — 0.86
- Least Common Multiple — 0.84
- Wilf Equivalence — 0.84
Computed from structural-signature embeddings · 2026-10-08