A090081 Cube root-smooth numbers: numbers k whose largest prime factor does not exceed the cube root of k.

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

Crossrefs

Cf. A063763, A001248, A048098, A036966, A054744, A059172, A068936, A076405, A076467, A006530, A175475.

Sequence in context: A341611 A083419 A329134 * A059172 A360558 A107606

Adjacent sequences: A090078 A090079 A090080 * A090082 A090083 A090084

Keyword

nonn,nice

Author

Labos Elemer, Nov 21 2003

Status

approved