The cuban primes, named after differences between successive cubic numbers, have the form . The first few are 7, 19, 37, 61, 127, 271, ... (Sloane's A002407), which are also the prime hex numbers. They correspond to indices , 3, 4, 5, 7, 10, 11, 12, 14, 15, 18, 24, 25, ... (Sloane's A002504; Cunningham 1912).

The numbers of cuban primes less than 1, 10, , ... are 0, 1, 4, 11, 28, 64, 173, 438, 1200, ... (Sloane's 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).

This entry contributed by Ed Pegg, Jr. (author's link)

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."

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Carmody, Phil; Pegg, Ed Jr.; and Weisstein, Eric W. "Cuban Prime." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/CubanPrime.html