Superabundant number

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:

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

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 LeonidasAlaoglu 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

Euler diagram of numbers under 100:

   Abundant

   Primitive abundant

   Highly abundant

  Superabundant and highly composite

   Colossally abundant and superior highly composite

   Weird

   Perfect

   Composite

   Deficient

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

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

where p_i is the i-th prime number, and

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

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 ${\displaystyle p_{k}}$ are factors of n. Then in particular any superabundant number is an even integer, and it is a multiple of the k-th primorial ${\displaystyle p_{k}\#.}$

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

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

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

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