Superabundant number

Class of natural numbers

In mathematics, a superabundant number is a certain kind of natural number. A natural number n is called superabundant precisely when, for all m < n:

{\frac {\sigma (m)}{m}}<{\frac {\sigma (n)}{n}}

where σ denotes the sum-of-divisors function (i.e., the sum of all positive divisors of n, including n itself). The first few superabundant numbers are 1, 2, 4, 6, 12, 24, 36, 48, 60, 120, ... (sequence A004394 in the OEIS). For example, the number 5 is not a superabundant number because for 1, 2, 3, 4, and 5, the sigma is 1, 3, 4, 7, 6, and 7/4 > 6/5.

Superabundant numbers were defined by Leonidas Alaoglu and Paul Erdős (1944). Unknown to Alaoglu and Erdős, about 30 pages of Ramanujan's 1915 paper "Highly Composite Numbers" were suppressed. Those pages were finally published in The Ramanujan Journal 1 (1997), 119–153. In section 59 of that paper, Ramanujan defines generalized highly composite numbers, which include the superabundant numbers.

Properties

Leonidas Alaoglu and Paul Erdős (1944) proved that if n is superabundant, then there exist a k and a₁, a₂, ..., a_k such that

n=\prod _{i=1}^{k}(p_{i})^{a_{i}}

where p_i is the i-th prime number, and

a_{1}\geq a_{2}\geq \dotsb \geq a_{k}\geq 1.

That is, they proved that if n is superabundant, the prime decomposition of n has non-increasing exponents (the exponent of a larger prime is never more than that a smaller prime) and that all primes up to $p_{k}$ are factors of n. Then in particular any superabundant number is an even integer, and it is a multiple of the k-th primorial $p_{k}\#.$

In fact, the last exponent a_k is equal to 1 except when n is 4 or 36.

Superabundant numbers are closely related to highly composite numbers. Not all superabundant numbers are highly composite numbers. In fact, only 449 superabundant and highly composite numbers are the same (sequence A166981 in the OEIS). For instance, 7560 is highly composite but not superabundant. Conversely, 1163962800 is superabundant but not highly composite.

Alaoglu and Erdős observed that all superabundant numbers are highly abundant.

Not all superabundant numbers are Harshad numbers. The first exception is the 105th superabundant number, 149602080797769600. The digit sum is 81, but 81 does not divide evenly into this superabundant number.

Superabundant numbers are also of interest in connection with the Riemann hypothesis, and with Robin's theorem that the Riemann hypothesis is equivalent to the statement that

{\frac {\sigma (n)}{e^{\gamma }n\log \log n}}<1

for all n greater than the largest known exception, the superabundant number 5040. If this inequality has a larger counterexample, proving the Riemann hypothesis to be false, the smallest such counterexample must be a superabundant number (Akbary & Friggstad 2009).

Not all superabundant numbers are colossally abundant.

Extension

The generalized $k$ -super abundant numbers are those such that ${\frac {\sigma _{k}(m)}{m^{k}}}<{\frac {\sigma _{k}(n)}{n^{k}}}$ for all $m<n$ , where $\sigma _{k}(n)$ is the sum of the $k$ -th powers of the divisors of $n$ .

1-super abundant numbers are superabundant numbers. 0-super abundant numbers are highly composite numbers.

For example, generalized 2-super abundant numbers are 1, 2, 4, 6, 12, 24, 48, 60, 120, 240, ... (sequence A208767 in the OEIS)

References

Briggs, Keith (2006), "Abundant numbers and the Riemann hypothesis", Experimental Mathematics, 15 (2): 251–256, doi:10.1080/10586458.2006.10128957, S2CID 46047029.
Akbary, Amir; Friggstad, Zachary (2009), "Superabundant numbers and the Riemann hypothesis", American Mathematical Monthly, 116 (3): 273–275, doi:10.4169/193009709X470128.
Alaoglu, Leonidas; Erdős, Paul (1944), "On highly composite and similar numbers", Transactions of the American Mathematical Society, 56 (3), American Mathematical Society: 448–469, doi:10.2307/1990319, JSTOR 1990319.

External links

MathWorld: Superabundant number

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