superabundant number

Noun

 * 1)  A positive integer whose abundancy index is greater than that of any lesser positive integer.
 * 2) * 1984, Richard K. Guy (editor), Reviews in Number Theory 1973-83, Volume 4, Part 1, As printed in Mathematical Reviews,, page 173,
 * The authors prove a theorem: If $$Q(X)$$ is the number of superabundant numbers $$\leqq X$$, then for $$c<5/48,\ Q(X) \geqq (\log X)^{1+c}$$ for sufficiently large $$X$$.
 * 1) * 1995, József Sándor, Dragoslav S. Mitrinović, Borislav Crstici, Handbook of Number Theory I, Springer, page 111,
 * 1) A number $$n$$ is called superabundant if $$\frac{\sigma(n)}{n}>\frac{\sigma(m)}{m}$$ for all $$m$$ with $$1\le m<n$$. Let $$Q(x)$$ be the counting function of superabundant numbers. Then:
 * a) If $$n$$ and $$n'$$ are two consecutive superabundant numbers then
 * $$\frac{n}{n'}<1+c(\log\log n)^2 / \log n$$
 * Corollary . $$Q(x)\ge c\log x\log\log x/(\log\log\log x)^2$$.

Usage notes

 * In mathematical terms, a positive integer $$n$$ is a superabundant number if $$\frac{\sigma(m)}{m}< \frac{\sigma(n)}{n}$$ for all positive integers $$m<n$$ (where $$\sigma(n)$$ denotes the sum of the divisors of $$n$$).

Translations

 * Finnish: ylirunsas luku
 * French: nombre superabondant
 * Romanian:
 * Russian: