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.
Scope of Application¶
Use the primitive label only after both semiperfectness and divisibility minimality have been checked.
- 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¶
Show a sum of distinct proper divisors equal to n, then rule out every smaller semiperfect divisor of n. Twenty passes both tests; twelve has a subset-sum witness but is divisible by semiperfect six and fails. Primitive here does not mean prime, and primary pseudoperfect uses a different reciprocal-sum equation. The OEIS list records minimal generators rather than all semiperfect multiples.
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.
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