Semiperfect number

Number equal to the sum of some of its divisors

A005835
Pseudoperfect (or semiperfect) numbers

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 the OEIS)

Properties

Every multiple of a semiperfect number is semiperfect.^[1] A semiperfect number that is not divisible by any smaller semiperfect number is called primitive.
Every number of the form 2^mp for a natural number m and an odd prime number p such that p < 2^m+1 is also semiperfect.
- In particular, every number of the form 2^m(2^m+1 − 1) is semiperfect, and indeed perfect if 2^m+1 − 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 the 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]

Every semiperfect number is a multiple of a primitive semiperfect number.

Notes

^ ^a ^b ^c Zachariou+Zachariou (1972)
^ ^a ^b ^c ^d 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. Nouvelle Série (in French). 2 (17): 212–213. MR 0199147. Zbl 0161.04402.
Zachariou, Andreas; Zachariou, Eleni (1972). "Perfect, semiperfect and Ore numbers". Bull. Soc. Math. Grèce. Nouvelle Série. 13: 12–22. MR 0360455. Zbl 0266.10012.

External links

Weisstein, Eric W. "Pseudoperfect Number". MathWorld.
Weisstein, Eric W. "Primitive semiperfect number". MathWorld.

Divisibility-based sets of integers

Overview

Factorization forms

Constrained divisor sums

With many divisors

Aliquot sequence-related

Base-dependent

Other sets

Classes of natural numbers

Powers and related numbers
Achilles Power of 2 Power of 3 Power of 10 Square Cube Fourth power Fifth power Sixth power Seventh power Eighth power Perfect power Powerful Prime power

Of the form a × 2^b ± 1
Cullen Double Mersenne Fermat Mersenne Proth Thabit Woodall

Other polynomial numbers
Hilbert Idoneal Leyland Loeschian Lucky numbers of Euler

Recursively defined numbers
Fibonacci Jacobsthal Leonardo Lucas Narayana Padovan Pell Perrin

Possessing a specific set of other numbers
Amenable Congruent Knödel Riesel Sierpiński

Expressible via specific sums
Nonhypotenuse Polite Practical Primary pseudoperfect Ulam Wolstenholme

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 Cubic Octahedral Dodecahedral Icosahedral Stella octangula
pyramidal	Square pyramidal

4-dimensional

non-centered	Pentatope Squared triangular Tesseractic

Combinatorial numbers
Bell Cake Catalan Dedekind Delannoy Euler Eulerian Fuss–Catalan Lah Lazy caterer's sequence Lobb Motzkin Narayana Ordered Bell Schröder Schröder–Hipparchus Stirling first Stirling second Telephone number Wedderburn–Etherington

Primes
Wieferich Wall–Sun–Sun Wolstenholme prime Wilson

Pseudoprimes
Carmichael number Catalan pseudoprime Elliptic pseudoprime Euler pseudoprime Euler–Jacobi pseudoprime Fermat pseudoprime Frobenius pseudoprime Lucas pseudoprime Lucas–Carmichael number Perrin pseudoprime Somer–Lucas pseudoprime Strong pseudoprime

Arithmetic functions and dynamics

Divisor functions	Abundant Almost perfect Arithmetic Betrothed Colossally abundant Deficient Descartes Hemiperfect Highly abundant Highly composite Hyperperfect Multiply perfect Perfect Practical Primitive abundant Quasiperfect Refactorable Semiperfect Sublime Superabundant Superior highly composite Superperfect
Prime omega functions	Almost prime Semiprime
Euler's totient function	Highly cototient Highly totient Noncototient Nontotient Perfect totient Sparsely totient
Aliquot sequences	Amicable Perfect Sociable Untouchable
Primorial	Euclid Fortunate

Other prime factor or divisor related numbers
Blum Cyclic Erdős–Nicolas Erdős–Woods Friendly Giuga Harmonic divisor Jordan–Pólya Lucas–Carmichael Pronic Regular Rough Smooth Sphenic Størmer Super-Poulet Zeisel

Numeral system-dependent numbers

Arithmetic functions
and dynamics

Persistence
- Additive
- Multiplicative

Digit sum	Digit sum Digital root Self Sum-product
Digit product	Multiplicative digital root Sum-product
Coding-related	Meertens
Other	Dudeney Factorion Kaprekar Kaprekar's constant Keith Lychrel Narcissistic Perfect digit-to-digit invariant Perfect digital invariant Happy