Semiperfect number

Demonstration, with Cuisenaire rods, of the perfection of the number 6

In number theory, a semiperfect number or pseudoperfect number is a natural number n that is equal to the sum of all or some of its proper divisors. A semiperfect number that is equal to the sum of all its proper divisors is a perfect number.

The first few semiperfect numbers are

6, 12, 18, 20, 24, 28, 30, 36, 40, ... (sequence A005835 in OEIS)

Properties

Every multiple of a semiperfect number is semiperfect.^[1] A semiperfect number that is not divisible by any smaller semiperfect number is primitive.
Every number of the form 2^mp for a natural number m and a prime number p such that p < 2^{m + 1} is also semiperfect.
- In particular, every number of the form 2^m − 1(2^m − 1) is semiperfect, and indeed perfect if 2^m − 1 is a Mersenne prime.
The smallest odd semiperfect number is 945 (see, e.g., Friedman 1993).
A semiperfect number is necessarily either perfect or abundant. An abundant number that is not semiperfect is called a weird number.
With the exception of 2, all primary pseudoperfect numbers are semiperfect.
Every practical number that is not a power of two is semiperfect.
The natural density of the set of semiperfect numbers exists.^[2]

Primitive semiperfect numbers

A primitive semiperfect number (also called a primitive pseudoperfect number, irreducible semiperfect number or irreducible pseudoperfect number) is a semiperfect number that has no semiperfect proper divisor.^[2]

The first few primitive semiperfect numbers are 6, 20, 28, 88, 104, 272, 304, 350, ... (sequence A006036 in OEIS)

There are infinitely many such numbers. All numbers of the form 2^mp, with p a prime between 2^m and 2^m+1, are primitive semiperfect, but this is not the only form: for example, 770.^[1]^[2] There are infinitely many odd primitive semiperfect numbers, the smallest being 945, a result of Paul Erdős:^[2] there are also infinitely many primitive semiperfect numbers that are not harmonic divisor numbers.^[1]

Notes

1 2 3 Zachariou+Zachariou (1972)
1 2 3 4 Guy (2004) p. 75

References

Friedman, Charles N. (1993). "Sums of divisors and Egyptian fractions". Journal of Number Theory 44 (3): 328–339. doi:10.1006/jnth.1993.1057. MR 1233293. Zbl 0781.11015.
Guy, Richard K. (2004). Unsolved Problems in Number Theory. Springer-Verlag. ISBN 0-387-20860-7. OCLC 54611248. Zbl 1058.11001. Section B2.
Sierpiński, Wacław (1965). "Sur les nombres pseudoparfaits". Mat. Vesn., N. Ser. 2 (in French) 17: 212–213. MR 199147. Zbl 0161.04402.
Zachariou, Andreas; Zachariou, Eleni (1972). "Perfect, semiperfect and Ore numbers". Bull. Soc. Math. Grèce, n. Ser. 13: 12–22. MR 360455. Zbl 0266.10012.

External links

Divisibility-based sets of integers

Overview	Integer factorization Divisor Unitary divisor Divisor function Prime factor Fundamental theorem of arithmetic Arithmetic number

Factorization forms	Prime Composite Semiprime Pronic Sphenic Square-free Powerful Perfect power Achilles Smooth Regular Rough Unusual

Constrained divisor sums	Perfect Almost perfect Quasiperfect Multiply perfect Hemiperfect Hyperperfect Superperfect Unitary perfect Semiperfect Practical Erdős–Nicolas

With many divisors	Abundant Primitive abundant Highly abundant Superabundant Colossally abundant Highly composite Superior highly composite Weird

Aliquot sequence-related	Untouchable Amicable Sociable Betrothed

Other sets	Deficient Friendly Solitary Sublime Harmonic divisor Frugal Equidigital Extravagant

Classes of natural numbers

Powers and
related numbers

Of the form
a × 2^b ± 1

Other polynomial
numbers

Recursively defined
numbers

Possessing a
specific set
of other numbers

Expressible via
specific sums

Generated
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Lucky

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Meertens

Figurate numbers

2-dimensional

centered	Centered triangular Centered square Centered pentagonal Centered hexagonal Centered heptagonal Centered octagonal Centered nonagonal Centered decagonal Star

non-centered	Triangular Square Square triangular Pentagonal Hexagonal Heptagonal Octagonal Nonagonal Decagonal Dodecagonal

3-dimensional

centered	Centered tetrahedral Centered cube Centered octahedral Centered dodecahedral Centered icosahedral

non-centered	Tetrahedral Octahedral Dodecahedral Icosahedral Stella octangula

pyramidal	Square pyramidal Pentagonal pyramidal Hexagonal pyramidal Heptagonal pyramidal

4-dimensional

centered	Centered pentachoric Squared triangular

non-centered	Pentatope

Pseudoprimes

Combinatorial
numbers

Arithmetic functions

By properties of σ(n)	Abundant Almost perfect Arithmetic Colossally abundant Descartes Hemiperfect Highly abundant Highly composite Hyperperfect Multiply perfect Perfect Practical number Primitive abundant Quasiperfect Refactorable Sublime Superabundant Superior highly composite Superperfect

By properties of Ω(n)	Almost prime Semiprime

By properties of φ(n)	Highly cototient Highly totient Noncototient Nontotient Perfect totient Sparsely totient

By properties of s(n)	Amicable Betrothed Deficient Semiperfect

Dividing a quotient

Other prime factor
or divisor related
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Recreational
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Base-dependent
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