A cuban prime is a prime number of the form 
for a positive
integer 
. The name refers to the difference between successive
cubic numbers. 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).
Cuban primes arise by evaluating the third homogeneous cyclotomic polynomial 
at 
. Every prime
factor of this value is congruent to 1 modulo 6.
In particular, 2 and 3 cannot be factors. Heuristically,
these restrictions favor prime number values compared
with unrestricted integers of the same size (P. Carmody,
pers. comm., Jan. 8, 2006).
