A number is said to be cubefree if its prime factorization contains no tripled factors. All primes are therefore trivially cubefree. The cubefree numbers are 1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, ... (OEIS A004709). The cubeful numbers (i.e., those that contain at least one cube) are 8, 16, 24, 27, 32, 40, 48, 54, ... (OEIS A046099). The number of cubefree numbers less than 10, 100, 1000, ... are 9, 85, 833, 8319, 83190, 831910, ..., and their asymptotic density is , where is the Riemann zeta function.

# Cubefree

## See also

Biquadratefree, Cubefree Part, Cubefree Word, Prime Number, Riemann Zeta Function, Squarefree## Explore with Wolfram|Alpha

## References

Sloane, N. J. A. Sequences A004709 and A046099 in "The On-Line Encyclopedia of Integer Sequences."## Referenced on Wolfram|Alpha

Cubefree## Cite this as:

Weisstein, Eric W. "Cubefree." From *MathWorld*--A Wolfram Web Resource. https://mathworld.wolfram.com/Cubefree.html