In math , a primitive abundant number is a special kind of abundant number . Its proper divisors , however, must all be deficient numbers(numbers whose sum of proper divisors are less than 2 times that number).[1] [2]
For example, 20 is a primitive abundant number because:
20 is an abundant number . This is because the sum of its divisors is 1 + 2 + 4 + 5 + 10 + 20 > 40. This makes 20 an abundant number .
The sums of the proper divisors of 1, 2, 4, 5 and 10 are 0, 1, 3, 1 and 8. All of these numbers are a deficient number. This makes 20 a primitive abundant number. The first few primitive abundant numbers are 20 , 70 , 88 , 104 , 272, 304, 368, 464, 550, 572 ... (sequence A071395 in the OEIS )
The smallest odd primitive abundant number is 945.
Every multiple of a primitive abundant number is abundant .
Every abundant number is a multiple of either a primitive abundant number or a multiple of a perfect number .
Every primitive abundant number is either a primitive semiperfect number or a weird number.
There is an infinite amount of primitive abundant numbers.[3]
↑ Eric W. Weisstein, Primitive Abundant Number at MathWorld.
↑ Erdős adopts a wider definition that requires a primitive abundant number to be not deficient, but not necessarily abundant (Erdős, Surányi and Guiduli. Topics in the Theory of Numbers p214. Springer 2003.). The Erdős definition allows perfect numbers to be primitive abundant numbers too.
↑ Paul Erdős, Journal of the London Mathematical Society 9 (1934) 278–282.
Other polynomial numbers
Hilbert
Idoneal
Leyland
Loeschian
Lucky numbers of Euler
Fibonacci
Jacobsthal
Leonardo
Lucas
Padovan
Pell
Perrin
Possessing a specific set of other numbers
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
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
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
Somer–Lucas pseudoprime
Strong pseudoprime
Arithmetic functions and dynamics
Divisor functions Prime omega functions Euler's totient function
Highly cototient
Highly totient
Noncototient
Nontotient
Perfect totient
Sparsely totient
Aliquot sequences
Amicable
Perfect
Sociable
Untouchable
Primorial
Blum
Cyclic
Erdős–Nicolas
Erdős–Woods
Friendly
Giuga
Harmonic divisor
Lucas–Carmichael
Pronic
Regular
Rough
Smooth
Sphenic
Størmer
Super-Poulet
Zeisel
Arithmetic functions and dynamics Digit sum
Digit sum
Digital root
Self
Sum-product
Digit product
Multiplicative digital root
Sum-product
Coding-related Other
Dudeney
Factorion
Kaprekar
Kaprekar's constant
Keith
Lychrel
Narcissistic
Perfect digit-to-digit invariant
Perfect digital invariant
P-adic numbers -relatedDigit -composition related
Palindromic
Pandigital
Repdigit
Repunit
Self-descriptive
Smarandache–Wellin
Strictly non-palindromic
Undulating
Digit-permutation related
Cyclic
Digit-reassembly
Parasitic
Primeval
Transposable
Divisor-related
Equidigital
Extravagant
Frugal
Harshad
Polydivisible
Smith
Vampire
Other
Pancake number
Sorting number
Divisibility-based sets of integers
Overview Factorization forms Constrained divisor sums
Perfect
Almost perfect
Quasiperfect
Multiply perfect
Hemiperfect
Hyperperfect
Superperfect
Unitary perfect
Semiperfect
Practical
Erdős–Nicolas
With many divisors Aliquot sequence-related
Untouchable
Amicable (Triple)
Sociable
Betrothed
Base -dependent
Equidigital
Extravagant
Frugal
Harshad
Polydivisible
Smith
Other sets
Arithmetic
Deficient
Friendly
Solitary
Sublime
Harmonic divisor
Descartes
Refactorable
Superperfect