A number is said to be biquadratefree (or quarticfree) if its prime factorization contains no quadrupled factors. All primes and prime powers with are therefore trivially biquadratefree. The biquadratefree numbers are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, ... (OEIS A046100). The biquadrateful numbers (i.e., those that contain at least one biquadrate) are 16, 32, 48, 64, 80, 81, 96, ... (OEIS A046101). The number of biquadratefree numbers less than 10, 100, 1000, ... are 10, 93, 925, 9240, 92395, 923939, ..., and their asymptotic density is , where is the Riemann zeta function.

# Biquadratefree

## See also

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

## References

Sloane, N. J. A. Sequences A046100 and A046101 in "The On-Line Encyclopedia of Integer Sequences."## Referenced on Wolfram|Alpha

Biquadratefree## Cite this as:

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