# Dodecahedral number

A dodecahedral number is a figurate number that represents a dodecahedron. The nth dodecahedral number is given by the formula

${\displaystyle {n(3n-1)(3n-2) \over 2}}$

The first such numbers are 0, 1, 20, 84, 220, 455, 816, 1330, 2024, 2925, 4060, 5456, 7140, 9139, 11480, … (sequence A006566 in the OEIS).

### Primality

A dodecahedral number can never be prime--the nth dodecahedral number is always divisible by n. This can be proved very simply:

• n is part of the numerator. There are no fractions in the numerator alone, so the numerator is divisible by n.
• Out of ${\displaystyle (3n-1)}$or ${\displaystyle (3n-2)}$, one of the two must be even. Therefore, the numerator is divisible by 2.
• Given the above, the numerator must be divisible by 2n.
• Noting the denominator, ${\displaystyle {2n \over 2}=n}$. Therefore, the nth dodecahedral number is always divisible by n.

## References

Kim, Hyun Kwang, On Regular Polytope Numbers (PDF), archived from the original (PDF) on 2010-03-07