Superabundant number
Encyclopedia
In mathematics
, a superabundant number (sometimes abbreviated as SA) is a certain kind of natural number
. Formally, a natural number n is called superabundant precisely when, for any m < 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
, ... .
Superabundant numbers were defined by . Unknown to Alaoglu and Erdos, 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.
where pl is the l-th prime number, and
Or to wit, if n is superabundant then the prime decomposition of n has decreasing exponents, the smaller
the prime, the larger its exponent.
In fact, ai is equal to 1 except when n is 4 or 36.
Superabundant numbers are closely related to highly composite number
s. It is false that all superabundant numbers are highly composite numbers. In fact, only 449 superabundant and highly composite numbers are the same. For instance, 7560 is highly composite but not superabundant. Alaoglu and Erdős observed that all superabundant numbers are highly abundant
. It is false that all superabundant numbers are Harshad number
s. The first exception is the 105th SA number, 149602080797769600. The digit sum is 81, but 81 does not divide evenly into this SA 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
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 .
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 superabundant number (sometimes abbreviated as SA) is a certain kind of 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...
. Formally, a natural number n is called superabundant precisely when, for any 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...
(i.e., the sum of all positive divisors of n, including n itself). The first few superabundant numbers are 1, 2, 4, 6, 12
12 (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...
, 24
24 (number)
24 is the natural number following 23 and preceding 25.The SI prefix for 1024 is yotta , and for 10−24 yocto...
, 36
36 (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...
, 48
48 (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...
, 60
60 (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:...
, 120
120 (number)
120 is the natural number following 119 and preceding 121. 120 was known as "the great hundred", especially prior to the year 1700, from the Teutonic Hundert which equalled 120. The number 100, now known commonly as "one hundred" was then known as "the small hundred". It is also known as...
, ... .
Superabundant numbers were defined by . Unknown to Alaoglu and Erdos, 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
proved that if n is superabundant, then there exist an i and a1, a2, ..., ai such thatwhere pl is the l-th prime number, and
Or to wit, if n is superabundant then the prime decomposition of n has decreasing exponents, the smaller
the prime, the larger its exponent.
In fact, ai is equal to 1 except when n is 4 or 36.
Superabundant numbers are closely related to highly composite number
Highly composite number
A highly composite number is a positive integer with more divisors than any positive integer smaller than itself.The initial or smallest twenty-one highly composite numbers are listed in the table at right....
s. It is false that all superabundant numbers are highly composite numbers. In fact, only 449 superabundant and highly composite numbers are the same. For instance, 7560 is highly composite but not superabundant. Alaoglu and Erdős observed that all superabundant numbers are highly abundant
Highly abundant number
In mathematics, a highly abundant number is a natural number with the property that the sum of its divisors is greater than the sum of the divisors of any smaller natural number....
. It is false that all superabundant numbers are Harshad number
Harshad number
A Harshad number, or Niven number in a given number base, is an integer that is divisible by the sum of its digits when written in that base. Harshad numbers were defined by D. R. Kaprekar, a mathematician from India. The word "Harshad" comes from the Sanskrit + , meaning joy-giver. The Niven...
s. The first exception is the 105th SA number, 149602080797769600. The digit sum is 81, but 81 does not divide evenly into this SA number.
Superabundant numbers are also of interest in connection with the Riemann hypothesis
Riemann hypothesis
In mathematics, the Riemann hypothesis, proposed by , is a conjecture about the location of the zeros of the Riemann zeta function which states that all non-trivial zeros have real part 1/2...
, and with Robin's theorem that the Riemann hypothesis is equivalent to the statement that
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 .