The cuban primes, named after differences between successive cubic numbers, have the form 
.
The first few are 7, 19, 37, 61, 127, 271, ... (OEIS A002407),
which are also the prime hex numbers. They correspond
to indices 
,
3, 4, 5, 7, 10, 11, 12, 14, 15, 18, 24, 25, ... (OEIS A002504;
Cunningham 1912).
The numbers of cuban primes less than 1, 10, 
, ... are 0, 1, 4, 11, 28, 64, 173, 438, 1200, ... (OEIS
A113478), which is well-approximated by
![ln[pi_c(x)]=lnx-0.8.](/images/equations/CubanPrime/NumberedEquation1.svg) |
Cuban primes are cyclotomic in nature, being the evaluation of the third homogeneous cyclotomic polynomial, 
, at values 
and 
. The form therefore can only have primitive factors of the
form 
. Also, by construction, 2 and 3 are
excluded as non-primitive factors. Therefore, this form has a slightly higher density
than would arbitrary numbers of the same size (P. Carmody, pers. comm., Jan. 8,
2006).
