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A090081
Cube root-smooth numbers: numbers k whose largest prime factor does not exceed the cube root of k.
3
1, 8, 16, 27, 32, 36, 48, 54, 64, 72, 81, 96, 108, 125, 128, 135, 144, 150, 160, 162, 180, 192, 200, 216, 225, 240, 243, 250, 256, 270, 288, 300, 320, 324, 343, 350, 360, 375, 378, 384, 392, 400, 405, 420, 432, 441, 448, 450, 480, 486, 490, 500, 504, 512, 525
OFFSET
1,2
COMMENTS
What is the asymptotic growth of this sequence?
Answer: a(n) ~ kn, where k = 1/A175475. That is, about 4.8% of numbers are in this sequence. - Charles R Greathouse IV, Jul 14 2014
LINKS
Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
FORMULA
Solutions to A006530(n) <= n^(1/3).
EXAMPLE
378 = 2 * 3^3 * 7 is a term of the sequence since 7 < 7.23... = 378^(1/3).
MAPLE
filter:= n ->
evalb(max(seq(f[1], f=ifactors(n)[2]))^3 <= n):
select(filter, [$1..1000]); # Robert Israel, Jul 14 2014
MATHEMATICA
ffi[x_] := Flatten[FactorInteger[x]]; ba[x_] := Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}]; lf[x_] := Length[FactorInteger[x]]; ma[x_] := Max[ba[x]]; Do[If[ !Greater[ma[n], gy=n^(1/3)//N]&&!PrimeQ[n], Print[n(*, {gy, ma[n]}}*)]], {n, 1, 1000}]
Select[Range[1000], (FactorInteger[#][[-1, 1]])^3 <= # &] (* T. D. Noe, Sep 14 2011 *)
PROG
(PARI) is(n)=my(f=factor(n)[, 1]); f[#f]^3<=n \\ Charles R Greathouse IV, Sep 14 2011
(Python)
from sympy import primefactors
def ok(n):
if n==1 or max(primefactors(n))**3<=n: return True
else: return False
print([n for n in range(1, 1001) if ok(n)]) # Indranil Ghosh, Apr 23 2017
KEYWORD
nonn,nice
AUTHOR
Labos Elemer, Nov 21 2003
STATUS
approved