Giuga number
A Giuga number is a composite number n such that for each of its distinct prime factors pi we have 
, or equivalently such that for each of its distinct prime factors pi we have 
.
The Giuga numbers are named after the mathematician Giuseppe Giuga, and relate to his conjecture on primality.
Definitions
Alternative definition for a Giuga number due to Takashi Agoh is: a composite number n is a Giuga number if and only if the congruence
holds true, where B is a Bernoulli number and 
is Euler's totient function.
An equivalent formulation due to Giuseppe Giuga is: a composite number n is a Giuga number if and only if the congruence
and if and only if
All known Giuga numbers n in fact satisfy the stronger condition
Examples
The sequence of Giuga numbers begins
For example, 30 is a Giuga number since its prime factors are 2, 3 and 5, and we can verify that
Properties
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