Highly abundant number
Encyclopedia
In mathematics
, a highly abundant number is a natural number
with the property that the sum of its divisors (including itself) is greater than the sum of the divisors of any smaller natural number.
Highly abundant numbers and several similar classes of numbers were first introduced by Pillai
(1943), and early work on the subject was done by Alaoglu
and Erdős
(1944). Alaoglu and Erdős tabulated all highly abundant numbers up to 104, and showed that the number of highly abundant numbers less than any N is at least proportional to log2 N. They also proved that 7200 is the largest powerful
highly abundant number, and therefore the largest highly abundant number with odd sum of divisors.
for all natural numbers m < n,
where σ denotes the sum-of-divisors function
. The first few highly abundant numbers are
For instance, 5 is not highly abundant because σ(5) = 5+1 = 6 is smaller than σ(4) = 4 + 2 + 1 = 7, while 8 is highly abundant because σ(8) = 8 + 4 + 2 + 1 = 15 is larger than all previous values of σ.
s are highly abundant numbers, but this is incorrect.
but there is a smaller number with larger sum of divisors,
so 9! is not highly abundant.
Alaoglu and Erdős noted that all superabundant numbers are highly abundant, and asked whether there are infinitely many highly abundant numbers that are not superabundant. This question was answered affirmatively by Nicolas (1969).
Despite the terminology, not all highly abundant numbers are abundant numbers. In particular, none of the first seven highly abundant numbers is abundant.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a highly abundant number is a natural number
Natural number
In mathematics, the natural numbers are the ordinary whole numbers used for counting and ordering . These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively...
with the property that the sum of its divisors (including itself) is greater than the sum of the divisors of any smaller natural number.
Highly abundant numbers and several similar classes of numbers were first introduced by Pillai
Subbayya Sivasankaranarayana Pillai
Subbayya Sivasankaranarayana Pillai was an Indian mathematician, well known for his work in number theory. He was from Tamil Nadu....
(1943), and early work on the subject was done by Alaoglu
Leonidas Alaoglu
Leonidas Alaoglu was a Canadian-American mathematician, most famous for his widely-cited result called Alaoglu's theorem on the weak-star compactness of the closed unit ball in the dual of a normed space, also known as the Banach–Alaoglu theorem.- Life and work :Alaoglu was born in Red Deer,...
and Erdős
Paul Erdos
Paul Erdős was a Hungarian mathematician. Erdős published more papers than any other mathematician in history, working with hundreds of collaborators. He worked on problems in combinatorics, graph theory, number theory, classical analysis, approximation theory, set theory, and probability theory...
(1944). Alaoglu and Erdős tabulated all highly abundant numbers up to 104, and showed that the number of highly abundant numbers less than any N is at least proportional to log2 N. They also proved that 7200 is the largest powerful
Powerful number
A powerful number is a positive integer m such that for every prime number p dividing m, p2 also divides m. Equivalently, a powerful number is the product of a square and a cube, that is, a number m of the form m = a2b3, where a and b are positive integers. Powerful numbers are also known as...
highly abundant number, and therefore the largest highly abundant number with odd sum of divisors.
Formal definition and examples
Formally, a natural number n is called highly abundant if and only ifIf and only if
In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
for all natural numbers m < n,
where σ denotes the sum-of-divisors function
Divisor function
In mathematics, and specifically in number theory, a divisor function is an arithmetical function related to the divisors of an integer. When referred to as the divisor function, it counts the number of divisors of an integer. It appears in a number of remarkable identities, including relationships...
. The first few highly abundant numbers are
- 1, 2, 3, 4, 6, 8, 1010 (number)10 is an even natural number following 9 and preceding 11.-In mathematics:Ten is a composite number, its proper divisors being , and...
, 1212 (number)12 is the natural number following 11 and preceding 13.The word "twelve" is the largest number with a single-morpheme name in English. Etymology suggests that "twelve" arises from the Germanic compound twalif "two-leftover", so a literal translation would yield "two remaining [after having ten...
, 1616 (number)16 is the natural number following 15 and preceding 17. 16 is a composite number, and a square number, being 42 = 4 × 4. It is the smallest number with exactly five divisors, its proper divisors being , , and ....
, 1818 (number)18 is the natural number following 17 and preceding 19.In speech, the numbers 18 and 80 are sometimes confused. When carefully enunciated, they differ in which syllable is stressed: 18 vs 80 . However, in dates such as 1864, or when contrasting numbers in the teens, such as 17, 18, 19, the stress...
, 2020 (number)20 is the natural number following 19 and preceding 21. A group of twenty units may also be referred to as a score.-In mathematics:*20 is the basis for vigesimal number systems....
, 2424 (number)24 is the natural number following 23 and preceding 25.The SI prefix for 1024 is yotta , and for 10−24 yocto...
, 3030 (number)30 is the natural number following 29 and preceding 31.-In mathematics:30 is the sum of the first four squares, which makes it a square pyramidal number.It is a primorial and is the smallest Giuga number....
, 3636 (number)36 is the natural number following 35 and preceding 37.- In mathematics :36 is both the square of 6 and a triangular number, making it a square triangular number...
, 4242 (number)42 is the natural number immediately following 41 and directly preceding 43. The number has received considerable attention in popular culture as a result of its central appearance in The Hitchhiker's Guide to the Galaxy as the "Answer to the Ultimate Question of Life, the Universe, and...
, 4848 (number)48 is the natural number following 47 and preceding 49. It is one third of a gross or four dozens.- In mathematics :Forty-eight is a double factorial of 6, a highly composite number. Like all other multiples of 6, it is a semiperfect number. 48 is the second 17-gonal number.48 is in abundance...
, 6060 (number)60 is the natural number following 59 and preceding 61. Being three times twenty, 60 is called "three score" in some older literature.-In mathematics:...
, ... .
For instance, 5 is not highly abundant because σ(5) = 5+1 = 6 is smaller than σ(4) = 4 + 2 + 1 = 7, while 8 is highly abundant because σ(8) = 8 + 4 + 2 + 1 = 15 is larger than all previous values of σ.
Relations with other sets of numbers
Some sources report that all factorialFactorial
In mathematics, the factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n...
s are highly abundant numbers, but this is incorrect.
- σ(9!) = σ(362880) = 1481040,
but there is a smaller number with larger sum of divisors,
- σ(360360) = 1572480,
so 9! is not highly abundant.
Alaoglu and Erdős noted that all superabundant numbers are highly abundant, and asked whether there are infinitely many highly abundant numbers that are not superabundant. This question was answered affirmatively by Nicolas (1969).
Despite the terminology, not all highly abundant numbers are abundant numbers. In particular, none of the first seven highly abundant numbers is abundant.