Cubefree
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.
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