Cuban Prime

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

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).

REFERENCES:

Cunningham, A. J. C. "On Quasi-Mersennian Numbers." Mess. Math. 41, 119-146, 1912.

Sloane, N. J. A. Sequences A002407/M4363, A002504/M0522, and A113478) in "The On-Line Encyclopedia of Integer Sequences."