Numbers with relatively many and large divisors

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The numbers with relatively many and large divisors are very interesting numbers to use as a base for the positional representation of numbers.

Definition

The number $\scriptstyle n\,$ is in the sequence if and only if

$\sigma (n)\geq e^{\gamma }n\log \log n,\,$

where $\scriptstyle \gamma \,$ is the Euler-Mascheroni constant and sigma(n) is the sum of divisors of n.

This definition gives the sequence

A067698 Numbers with relatively many and large divisors (see comments).

{2, 3, 4, 5, 6, 8, 9, 10, 12, 16, 18, 20, 24, 30, 36, 48, 60, 72, 84, 120, 180, 240, 360, 720, 840, 2520, 5040}

Robin has shown that 5040 is the last element in the sequence if and only if the Riemann hypothesis is true. Moreover the sequence is infinite if the Riemann hypothesis is false. Gronwall's theorem says that

$\lim _{\sup }{\frac {\sigma (n)}{n\log \log n}}=e^{\gamma }\,$

See also

• Numbers with relatively many divisors

References

• Guy Robin, Grandes valeurs de la fonction somme des diviseurs et hypothèse de Riemann, J. Math. Pures Appl. 63 (1984), 187-213.