Giuga numbers

A composite number $n$

is called Giuga number if $p$

divides

for all prime divisors $p$

It can be proved that $n$ is a Giuga number if and only if the difference

$\sum_{p|n}\frac{1}{p}-\prod_{p|n}\frac{1}{p}\,,$

where

ranges among the divisors of $n$

, represents a natural number.

For example, $858=2\cdot3\cdot11\cdot13$ is a Giuga number because

$\frac{1}{2}+\frac{1}{3}+\frac{1}{11}+\frac{1}{13}- \frac{1}{2}\cdot\frac{1}{3}\cdot\frac{1}{11}\cdot\frac{1}{13}=1$

It can be proved easily that a Giuga number must be squarefree and equal to the product of at least 3 primes.

All the known Giuga numbers are even, but the possibility of an odd Giuga number has not been ruled out.

Pictorial representation of remainders (mod 2, 3, ...,11) frequency. For a table of values and more details click here

A graph displaying how many Giuga numbers are multiples of the primes p from 2 to 71. In black the ideal line 1/p.

Useful links Mathworld, Giuga Number
Wikipedia, Giuga number
OEIS, Sequence A007850

Giuga numbers can also be... (you may click on names or numbers)