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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 .

Scope of Application

  • 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.

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.

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.

Abstract Reasoning

  1. Type the carrier. Identify the number theory entities to which the claim applies.
  2. State the relation. Use the source-grounded identity: In number theory, a superperfect number is a positive integer that satisfies.
  3. Check operation and conditions. The m-superperfect numbers are in turn examples of (m,k)-perfect numbers which satisfy.
  4. 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.
  5. Test variation.

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.

Relationships to Other Abstractions

Local relationship map for Superperfect numberParents appear above the current abstraction, mutual partners to the right, and children below. Node labels state whether each abstraction is prime or domain-specific; colors identify relation types.Superperfect numberDOMAINPrime abstraction: Constraint — is a kind ofConstraintPRIME

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

Neighborhood in Abstraction Space

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

Family — Unclustered & Miscellaneous (2551 abstractions)

Nearest neighbors

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