Primitive Semiperfect Number¶
A semiperfect integer with no smaller semiperfect divisor.
Core Idea¶
A semiperfect (pseudoperfect) positive integer can be written as the sum of some distinct proper divisors. A primitive semiperfect number satisfies that condition but has no smaller semiperfect divisor. Primitive here means minimal under divisibility within the semiperfect set; it does not mean prime in the usual factorization sense, and it is unrelated to the reciprocal-sum definition of a primary pseudoperfect number.
Twenty supplies a compact test: 1+4+5+10=20, and none of its proper divisors is semiperfect. Twelve is the nearest contrast because a divisor subset can sum to 12, yet 6 is semiperfect and divides 12. The distinction matters because semiperfectness passes to multiples, so a catalog of primitive members identifies a generating basis rather than every member of the larger class. OEIS A006036 documents that basis by enumerating its initial terms.
Structural Signature¶
Sig role-phrases:
- Positive integer candidate — Supplies n and its positive proper divisors as the domain of the test. It is constitutive. Counterfactual: A negative integer or an arbitrary set is not this number class.
- Distinct proper-divisor subset — Witnesses semiperfectness when its members sum exactly to n. It is constitutive. Counterfactual: A sum using n itself or a repeated divisor is not a qualifying witness.
- Smaller semiperfect divisors — Provides the comparison set of positive d<n dividing n that are themselves semiperfect. It is constitutive. Counterfactual: Checking only one smaller integer can miss another semiperfect divisor.
- Divisibility-minimal condition — Requires that no member of the smaller semiperfect comparison set exists. It is constitutive. Counterfactual: A semiperfect multiple of 6 such as 12 fails primitiveness despite its own divisor-sum witness.
- Representation versus generator distinction — Separates the subset-sum witness of semiperfectness from the minimal-basis test across integers. It is boundary. Counterfactual: Showing n semiperfect alone does not establish primitive status.
What It Is Not¶
- Not prime number. A primitive semiperfect integer can be composite, and the adjective concerns divisibility within one special class.
- Not just semiperfect. A proper-divisor subset sum is necessary but does not rule out smaller semiperfect divisors.
- Not a perfect-number requirement. Some primitive members, such as 20, are abundant rather than perfect.
- Not primary pseudoperfect. That separate term uses a reciprocal sum over prime factors.
- Closest near-miss. Twelve is the closest near miss: it is semiperfect, for example 2+4+6=12, but 6 is a smaller semiperfect divisor of 12.
Scope of Application¶
- Integer-sequence classification. Separate divisibility-minimal semiperfect members from their multiples.
- Divisor-sum reasoning. Exhibit a proper-divisor subset witness before testing minimality.
- Catalog interpretation. Read OEIS A006036 as primitive members, not all pseudoperfect integers.
- Adjacent classes. Distinguish primitive semiperfect from perfect, abundant, weird, and primary pseudoperfect notions.
Clarity¶
First prove n semiperfect with a sum of distinct proper divisors. Then check whether any smaller divisor d of n is itself semiperfect. Twenty passes both tests; twelve fails the second because six divides it. A semiperfect number need not be perfect, and primitive is not synonymous with prime. A primary pseudoperfect reciprocal identity is a different property despite the similar name.
Manages Complexity¶
The adjective primitive compresses a two-level test: a subset sum inside n's divisor set and a minimality test across semiperfect divisors. It avoids listing every semiperfect multiple but can hide which witness and which excluded divisors support membership. Keeping both certificates separate makes the sequence auditable.
Abstract Reasoning¶
- List the positive proper divisors of candidate n.
- Find or rule out a distinct subset summing exactly to n.
- For a semiperfect n, inspect each smaller divisor for semiperfectness.
- Declare n primitive only if the smaller-semiperfect-divisor set is empty.
- Contrast a semiperfect multiple, such as 12 versus 6, and do not infer primality.
Knowledge Transfer¶
The two-stage witness/minimality test transfers to other integers under the same proper-divisor convention, and the set of primitive semiperfect members generates nonprimitive multiples because semiperfectness is multiple-closed. A witness for 20 cannot be copied unchanged to 28 or 88, and 'primitive' in group theory, algebra, or prime factorization has other tests. OEIS's catalog terms do not transfer numerical density claims without further proof.
Examples¶
Canonical¶
For n=20 the proper divisors 1, 2, 4, 5, and 10 include the distinct subset 1+4+5+10=20, so 20 is semiperfect. The smaller semiperfect numbers 6, 12, and 18 do not divide 20; in fact its proper divisors 1, 2, 4, 5, and 10 are not semiperfect. Thus 20 is primitive. In contrast, 12 has its own semiperfect witness but fails primitiveness because 6 divides it.
Mapped back: Positive integer candidate → n=20; Distinct proper-divisor subset → 1,4,5,10 sum to 20; Smaller semiperfect divisors → none among 20's proper divisors; Divisibility-minimal condition → 20 is not a semiperfect multiple of a smaller number; Representation versus generator distinction → 12 demonstrates subset-sum success without minimality.
Applied / In Practice¶
The On-Line Encyclopedia of Integer Sequences maintains A006036 as the primitive pseudoperfect-number sequence and lists 6, 20, 28, 88, and 104 at its start. This is an attested research-catalog use of the divisibility-minimal filter: the sequence records generators rather than every semiperfect multiple. Its listed terms do not by themselves prove a conjecture about all semiperfect numbers or supply an efficient membership algorithm.
Mapped back: Positive integer candidate → each positive integer tested for A006036; Distinct proper-divisor subset → pseudoperfect witness required by the sequence definition; Smaller semiperfect divisors → excluded by OEIS primitive criterion; Divisibility-minimal condition → sequence contains 6,20,28 rather than every multiple; Representation versus generator distinction → catalog of primitive basis members, not all semiperfect terms.
Structural Tensions¶
T1 — Subset-Sum Witness versus Divisor-Minimality. A semiperfect representation shows the first property but leaves the separate primitiveness question open.
Diagnostic: Has every smaller semiperfect divisor been excluded?
T2 — Primitive Basis versus Multiple Closure. Every multiple of a semiperfect number remains semiperfect, creating many nonprimitive members from a minimal generator.
Diagnostic: Is this number a new generator or inherited through divisibility?
Structural–Framed Character¶
The skeleton is minimal membership within a divisibility-ordered class. A primitive semiperfect number has a witness as a sum of distinct proper divisors and no smaller semiperfect proper divisor. It is an approved unparented root because the live graph lacks a semiperfect-number genus.
Evaluative weight: “Primitive” is a precise minimality condition, not praise for simplicity.
Human-practice-bound: Proper-divisor and distinct-subset conventions specify the test.
Institutional origin: Number theory defines the arithmetic membership and divisibility order.
Vocabulary travels: Primitive in groups or polynomials has a different criterion.
Import versus recognize: The witness-then-minimality test applies to new integers; one number’s divisor subset cannot simply be copied to another.
Its character: A specialized integer class, not a prime for minimal objects.
Structural Core vs. Domain Accent¶
Skeletal core. An object can satisfy a property while none of its smaller divisibility predecessors does.
Domain-bound accent. A positive integer is semiperfect if distinct proper divisors sum to it; it is primitive semiperfect if no semiperfect proper divisor remains below it.
Why not prime. Generic minimality lacks the proper-divisor subset-sum condition. “Primitive” in another algebraic context is only a lexical neighbor.
Instantiates / Related Primes¶
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Related — semiperfect number. Every primitive member is semiperfect; many semiperfect multiples are not primitive.
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Related — perfect number. Six and 28 are perfect primitive semiperfect members, but twenty shows perfectness is not necessary.
Neighborhood in Abstraction Space¶
Primitive Semiperfect Number sits in a moderately populated region (43rd percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.
Family — Group Structure & Subgroup Properties (10 abstractions)
Nearest neighbors
- Achilles Number — 0.88
- Semiperfect Number — 0.88
- Number of groups of a given order — 0.86
- Sums of three cubes — 0.86
- Log-Sum Inequality — 0.86
Computed from structural-signature embeddings · 2026-10-08
Not to Be Confused With¶
- Semiperfect number. Tell: Was a smaller semiperfect divisor also excluded?
- Prime number. Tell: Does primitive refer to this class's divisibility order rather than factorization?
- Primary pseudoperfect number. Tell: Is the test a proper-divisor subset sum or a reciprocal prime-factor equation?
- Weird number. Tell: Does any proper-divisor subset sum to n at all?
References¶
- OEIS Foundation, A006036 Primitive pseudoperfect numbers: https://oeis.org/A006036
- Frozen Wikipedia discovery revision: https://en.wikipedia.org/wiki/Semiperfect_number (revision 1309480096).