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A positive integer $n$ is extremely abundant if either $n=10080$, or $n>10080$ and

$$σ(n)/(n×log(log (n)))>σ(m)/(m×log(log (m)))$$

for all $10080≤m<n$. Here $σ(n)$ is the sum-of-divisors function and $log$ is the natural logarithm.

My question is: About any known result relating prime numbers with extremely abundant numbers.

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  • $\begingroup$ Why $10080$? Anyway, these numbers are tabulated at oeis.org/A217867 where there are some links to the literature. Maybe you'll find something useful there, and then you can report back to us! $\endgroup$
    – Gerry Myerson
    Commented Nov 2, 2019 at 11:29
  • $\begingroup$ @GerryMyerson: But the only link is related to RH. $\endgroup$
    – Safwane
    Commented Nov 2, 2019 at 13:50
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    $\begingroup$ No, Germany, the link in Carlo's answer is also at that oeis page. $\endgroup$
    – Gerry Myerson
    Commented Nov 2, 2019 at 21:03
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    $\begingroup$ More recent question by same user on extremely abundant numbers: mathoverflow.net/questions/347446/… $\endgroup$
    – Gerry Myerson
    Commented Dec 2, 2019 at 21:49

1 Answer 1

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This paper lists several statements relating extremely abundant numbers and prime numbers, for example:

  • There is an infinite number of primes which cannot be the largest prime factor of any extremely abundant number.

  • The largest prime factor $p(n)$ of any extremely abundant number $n$ satisfies $p(n)<\log n$.

  • If $m\leq n$ then $p(m)\leq p(n)$.

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