Superperfect number¶

In number theory, a superperfect number is a positive integer that satisfies.

Version: v1 · 2026-09-28 · History
Domain-specific #: 12378
Domain group: Formal Sciences
Origin domain: Mathematics
Subdomain: Number Theory → Mathematics

Core Idea¶

Superperfect number is treated here as the recurring number theory identity summarized by this source-grounded definition: In number theory, a superperfect number is a positive integer that satisfies.

In number theory, a superperfect number is a positive integer that satisfies. \sigma^2(n)=\sigma(\sigma(n))=2n\, ,. Superperfect numbers are not a generalization of perfect numbers but have a common generalization.

2, 4, 16, 64, 4096, 65536, 262144, 1073741824, ... . To illustrate: it can be seen that 16 is a superperfect number as , and , thus . If is an even superperfect number, then must be a power of 2, , such that is a Mersenne prime.

For Superperfect number, the abstraction is narrower than the article's general subject matter: a positive case must preserve In number theory, a superperfect number is a positive integer that satisfies. Retaining only the name, a familiar example, or a downstream effect is insufficient. The specialist roles and tests remain anchored in number theory, which is why this identity is domain-specific rather than prime.

Structural Signature¶

Sig role-phrases:

Defining carrier — An odd superperfect number would have to be a square number such that either or is divisible by at least three distinct primes.
Constitutive relation — Perfect and superperfect numbers are examples of the wider class of m-superperfect numbers, which satisfy.
Operating condition — The m-superperfect numbers are in turn examples of (m,k)-perfect numbers which satisfy.
Recognition evidence — With this notation, perfect numbers are (1,2)-perfect, multiperfect numbers are (1,k)-perfect, superperfect numbers are (2,2)-perfect and m-superperfect numbers are (m,2)-perfect.
Admissible variation — corresponding to m = 1 and 2 respectively.
Characteristic consequence — For m ≥ 3 there are no even m-superperfect numbers.
Failure boundary — Examples of classes of (m,k)-perfect numbers are.

What It Is Not¶

Not the whole field of number theory. The node requires the specific identity stated by In number theory, a superperfect number is a positive integer that satisfies.
Not an over-broad reading. Superperfect numbers are not a generalization of perfect numbers but have a common generalization.
Not an over-broad reading. It is not known whether there are any odd superperfect numbers.
Not an over-broad reading. Perfect and superperfect numbers are examples of the wider class of m-superperfect numbers, which satisfy.
Not automatically Quasiperfect number. Retrieval proximity does not establish equivalence; the two identities must be compared by carrier, operation, and failure boundary.

Scope of Application¶

Superperfect number applies literally inside number theory wherever the source-defined carrier and relation can be established. Its documented habitats include:

Generalizations. Perfect and superperfect numbers are examples of the wider class of m-superperfect numbers, which satisfy.
Generalizations. The m-superperfect numbers are in turn examples of (m,k)-perfect numbers which satisfy.
Generalizations. With this notation, perfect numbers are (1,2)-perfect, multiperfect numbers are (1,k)-perfect, superperfect numbers are (2,2)-perfect and m-superperfect numbers are (m,2)-perfect.
Generalizations. corresponding to m = 1 and 2 respectively.
Generalizations. For m ≥ 3 there are no even m-superperfect numbers.
Generalizations. Examples of classes of (m,k)-perfect numbers are.

Outside number theory, the name should be retained only when these same operational conditions survive; otherwise the comparison belongs to the broader parent Theory or should be marked as analogy.

Clarity¶

A clear use of Superperfect number names the carrier, the operative relation, and the conditions under which the source treats the identity as present. The minimal definition is In number theory, a superperfect number is a positive integer that satisfies. The strongest recognition evidence in the frozen account is: With this notation, perfect numbers are (1,2)-perfect, multiperfect numbers are (1,k)-perfect, superperfect numbers are (2,2)-perfect and m-superperfect numbers are (m,2)-perfect. A report should distinguish that evidence from a proxy, consequence, or common implementation. It should also state the qualification Superperfect numbers are not a generalization of perfect numbers but have a common generalization. so that a reader can reproduce the classification rather than infer it from topical resemblance.

Manages Complexity¶

Superperfect number compresses multiple number theory details into a stable diagnostic relation. The source shows both the central mechanism—perfect and superperfect numbers are examples of the wider class of m-superperfect numbers, which satisfy.—and the practical consequence—for m ≥ 3 there are no even m-superperfect numbers. This compression makes cases comparable while leaving parameters, conventions, exceptions, and evidential quality explicit. It is lossy by design: local history and implementation details may be omitted only when they do not alter the defining relation.

Abstract Reasoning¶

Type the carrier. Identify the number theory entities to which the claim applies.
State the relation. Use the source-grounded identity: In number theory, a superperfect number is a positive integer that satisfies.
Check operation and conditions. The m-superperfect numbers are in turn examples of (m,k)-perfect numbers which satisfy.
Demand recognition evidence. With this notation, perfect numbers are (1,2)-perfect, multiperfect numbers are (1,k)-perfect, superperfect numbers are (2,2)-perfect and m-superperfect numbers are (m,2)-perfect.
Test variation. Change an implementation or setting while preserving corresponding to m = 1 and 2 respectively.
Run the collapse test. Remove the defining operation; if the label still seems equally apt, only a topic or correlate was retained.
Reduce cautiously. When the specialist conditions cannot be carried, route the residual comparison to Theory.

Knowledge Transfer¶

Within the home domain. Knowledge about Superperfect number transfers literally when a new case preserves the same carrier type, relation, and recognition test. Perfect and superperfect numbers are examples of the wider class of m-superperfect numbers, which satisfy. The m-superperfect numbers are in turn examples of (m,k)-perfect numbers which satisfy.

Beyond the home domain. No canonical parent is asserted for Superperfect number. An outside case receives the specialist name only when the same typed roles and rejection conditions can be filled literally; otherwise the comparison remains an analogy pending later graph densification.

Examples¶

Canonical¶

Perfect and superperfect numbers are examples of the wider class of m-superperfect numbers, which satisfy. This case is canonical because it supplies a concrete carrier and lets the defining relation be checked rather than merely named.

Mapped back: carrier → the entities in the documented case; operation → In number theory, a superperfect number is a positive integer that satisfies; recognition evidence → With this notation, perfect numbers are (1,2)-perfect, multiperfect numbers are (1,k)-perfect, superperfect numbers are (2,2)-perfect and m-superperfect numbers are (m,2)-perfect

Applied / In Practice¶

The m-superperfect numbers are in turn examples of (m,k)-perfect numbers which satisfy. The applied case shows how the identity is used under a second setting or qualification while keeping the same operative relation.

Mapped back: changed setting → Generalizations; invariant → In number theory, a superperfect number is a positive integer that satisfies; boundary → the case exits the class when superperfect numbers are not a generalization of perfect numbers but have a common generalization

Structural Tensions¶

T1 — Stable identity versus admissible variation. Superperfect numbers are not a generalization of perfect numbers but have a common generalization. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Which changes preserve the defining relation, and which replace it?

T2 — Recognition versus proxy. It is not known whether there are any odd superperfect numbers. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Does the cited evidence establish the identity or only a correlated sign?

T3 — Definition versus implementation. Perfect and superperfect numbers are examples of the wider class of m-superperfect numbers, which satisfy. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Is the observed implementation constitutive, optional, or merely common?

T4 — Scope versus overextension. The m-superperfect numbers are in turn examples of (m,k)-perfect numbers which satisfy. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Can every claimed application fill the same typed roles without metaphor?

T5 — Transfer versus domain accent. An odd superperfect number would have to be a square number such that either or is divisible by at least three distinct primes. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Does the receiving case instantiate Superperfect number literally, co-instantiate Theory, or only resemble it?

T6 — Autonomy versus reduction. Perfect and superperfect numbers are examples of the wider class of m-superperfect numbers, which satisfy. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: What does Superperfect number distinguish that the broader parent Theory leaves together?

Terminal boundary synthesis. For Superperfect number, the terminal identity test begins with the definition In number theory, a superperfect number is a positive integer that satisfies.. A reviewer must then establish the carrier and operation described by An odd superperfect number would have to be a square number such that either or is divisible by at least three distinct primes. and Perfect and superperfect numbers are examples of the wider class of m-superperfect numbers, which satisfy.. Recognition is constrained by The m-superperfect numbers are in turn examples of (m,k)-perfect numbers which satisfy., while admissible variation is limited by With this notation, perfect numbers are (1,2)-perfect, multiperfect numbers are (1,k)-perfect, superperfect numbers are (2,2)-perfect and m-superperfect numbers are (m,2)-perfect. and the collapse boundary corresponding to m = 1 and 2 respectively.. The source-domain setting in number theory matters because Perfect and superperfect numbers are examples of the wider class of m-superperfect numbers, which satisfy. and The m-superperfect numbers are in turn examples of (m,k)-perfect numbers which satisfy. specify where those roles have literal occupants. The strongest negative controls are The node requires the specific identity stated by In number theory, a superperfect number is a positive integer that satisfies. and Superperfect numbers are not a generalization of perfect numbers but have a common generalization.; a case satisfying either exclusion should not be rescued merely because its label or examples look familiar.

Terminal adjudication sequence. First, bind the claimed instance to a concrete carrier and state the criterion by which In number theory, a superperfect number is a positive integer that satisfies. is recognized. Second, vary implementation, scale, notation, and example while holding Perfect and superperfect numbers are examples of the wider class of m-superperfect numbers, which satisfy. fixed; persistence supports one identity rather than several topic fragments. Third, remove The m-superperfect numbers are in turn examples of (m,k)-perfect numbers which satisfy. or trigger corresponding to m = 1 and 2 respectively. and verify that the classification fails. Fourth, compare the result with the two negative controls instead of relying on name similarity. Fifth, check scope against Perfect and superperfect numbers are examples of the wider class of m-superperfect numbers, which satisfy. and record any qualification supplied by number theory. Finally, audit the graph claim. The approved unparented placement prevents a weak lexical resemblance from becoming a false ontological claim; a later edge must preserve every constitutive role stated here. This sequence makes the entry rejectable, keeps analogy separate from literal transfer, and exposes which fact would require revision.

Counterfactual boundary matrix. Evaluate Superperfect number under four controlled substitutions. In the carrier substitution, replace the concrete entities while retaining An odd superperfect number would have to be a square number such that either or is divisible by at least three distinct primes.; the identity should persist only if the new carrier has the same operative type. In the operation substitution, replace Perfect and superperfect numbers are examples of the wider class of m-superperfect numbers, which satisfy. while preserving surface vocabulary; the identity should fail unless the replacement entails the same relation. In the evidence substitution, change the instrument, representation, or witness used for The m-superperfect numbers are in turn examples of (m,k)-perfect numbers which satisfy.; classification may persist when the new evidence warrants the same fact. In the scope substitution, move the case outside Perfect and superperfect numbers are examples of the wider class of m-superperfect numbers, which satisfy. and ask whether The m-superperfect numbers are in turn examples of (m,k)-perfect numbers which satisfy. still gives the roles literal occupants. These four tests separate constitutive structure from implementation, evidence, and familiar examples. They also identify the exact revision needed when a source expands or narrows the recognized class.

Neighbor and residual test. The negative controls The node requires the specific identity stated by In number theory, a superperfect number is a positive integer that satisfies. and Superperfect numbers are not a generalization of perfect numbers but have a common generalization. define two directions of possible overreach. A reviewer should construct one case that satisfies the first control but not Superperfect number, one that satisfies Superperfect number but not the control, and the corresponding pair for the second control. If no such asymmetric pair can be stated, the candidate may duplicate a neighbor or the distinction may depend only on wording. When the specialist identity fails but a thinner relation remains, record that residual separately instead of stretching Superperfect number. The approved unparented placement prevents a weak lexical resemblance from becoming a false ontological claim; a later edge must preserve every constitutive role stated here. The resulting decision trail makes later DAG densification possible without treating today's uncertainty as a hierarchy fact.

Structural–Framed Character¶

Superperfect number is mixed or framed-leaning. Its structural side is the repeatable organization summarized by In number theory, a superperfect number is a positive integer that satisfies. Its framed side is the number theory vocabulary that fixes the carrier, evidence, exceptions, and admissible transformations.

Evaluative weight: the identity can be stated descriptively even when applications carry practical stakes. Human-practice dependence: the source-grounded carrier determines whether the relation exists independently or is constituted by a practice. Institutional origin: disciplinary conventions stabilize the name and test. Vocabulary portability: The m-superperfect numbers are in turn examples of (m,k)-perfect numbers which satisfy. Import versus recognition: literal transfer requires the same mechanism; shape alone is analogy.

Its portable skeleton is Theory. Its character: a recurring specialist identity whose thin organization can be abstracted, while its operational meaning remains domain-bound.

Structural Core vs. Domain Accent¶

What is skeletal. In number theory, a superperfect number is a positive integer that satisfies. The stable skeleton is the typed relation expressed in that definition and the entry's recognition and collapse tests. The source identifies these operative conditions: An odd superperfect number would have to be a square number such that either or is divisible by at least three distinct primes. Perfect and superperfect numbers are examples of the wider class of m-superperfect numbers, which satisfy. It further constrains recognition and variation through: The m-superperfect numbers are in turn examples of (m,k)-perfect numbers which satisfy. With this notation, perfect numbers are (1,2)-perfect, multiperfect numbers are (1,k)-perfect, superperfect numbers are (2,2)-perfect and m-superperfect numbers are (m,2)-perfect.

What is domain-bound. number theory supplies the operative entities, technical vocabulary, warrants, and exceptions that make Superperfect number literal. Its documented scope includes the condition that Perfect and superperfect numbers are examples of the wider class of m-superperfect numbers, which satisfy. Another bounded application condition is that The m-superperfect numbers are in turn examples of (m,k)-perfect numbers which satisfy. These are not decorative examples; they determine which carrier and evidence can fill the abstraction's roles.

Why no parent is asserted. Removing those specialist details does not currently yield one live catalog node that is a necessary genus for every instance. The entry is therefore approved as unparented rather than attached by topical resemblance. Its collapse evidence remains specific—corresponding to m = 1 and 2 respectively.—and future graph densification may discover a defensible relation only if it preserves that boundary.

This entry is a kind of Constraint.

Approved unparented node. No current live node supplies a defensible necessary genus or structural prerequisite for Superperfect number. The reviewed identity is: In number theory, a superperfect number is a positive integer that satisfies. The accelerated suggestion was declined because topical or lexical similarity does not establish hierarchy; the node is admitted without a parent pending later graph densification.
Related reasoning operations. Evidence, representation, comparison, classification, transformation, or evaluation may participate in particular cases, but participation does not make any one of them a necessary parent of every instance.

Relationships to Other Abstractions¶

Current abstraction Superperfect number Domain-specific

Parents (1) — more general patterns this builds on

Superperfect number is a kind of Constraint Prime

Superperfect number membership is the checkable iterated divisor-sum equation sigma(sigma(n)) = 2n, structurally the same pattern as perfect and hyperperfect numbers.

Hierarchy path (1) — routes to 1 parentless root

Superperfect number → Constraint

Neighborhood in Abstraction Space¶

Superperfect number sits in a sparse region of the domain-specific corpus (78^th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Unclustered & Miscellaneous (2551 abstractions)

Nearest neighbors

Binade — 0.84
Zero Divisor — 0.83
Square-Free Integer — 0.83
Typographical Number Theory — 0.83
Sublime number — 0.82

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

Not to Be Confused With¶

Theory. The parent omits the specialist differentia. Tell: Can the case establish In number theory, a superperfect number is a positive integer that satisfies?
Quasiperfect number. Classify a positive integer by the divisor-sum equation sigma of n equals twice n plus one, an unresolved class whose hypothetical members must satisfy strong odd-square and prime-factor restrictions. Tell: Which entry's carrier, operation, and failure condition are satisfied?
Hyperperfect number. Classify a natural number by the parameterized divisor-sum constraint n = 1 + k(σ(n) − n − 1), with perfect numbers as the k = 1 boundary case. Tell: Which entry's carrier, operation, and failure condition are satisfied?
Superpermutation. Construct a string over n symbols whose contiguous substrings include every permutation of those symbols, then minimize length by maximizing compatible overlap among required permutation windows. Tell: Which entry's carrier, operation, and failure condition are satisfied?
A measurement, proxy, or consequence. Those may provide evidence without being the identity. Tell: Would Superperfect number remain present if the detector or downstream effect changed?
A metaphorical analogue. A similar shape outside number theory lacks the specialist mechanism. Tell: Do the native roles transfer literally, or only the parent Theory?

References¶

Frozen Wikipedia discovery revision: https://en.wikipedia.org/wiki/Superperfect_number (revision 1365374288).
Preserved source candidate: https://ir.cwi.nl/pub/10355

The frozen Wikipedia revision is discovery provenance. The retained source set was reviewed for identity, formal or operational relation, and scope. The encyclopedia's structural synthesis is bounded to those claims; a thin authority surface is recorded as a nonblocking source-strengthening repair rather than concealed.