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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 (list; graph; refs; listen; history; text; internal format)
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 A107606 A354561

Adjacent sequences:  A090078 A090079 A090080 * A090082 A090083 A090084

KEYWORD

nonn,nice,changed

AUTHOR

Labos Elemer, Nov 21 2003

STATUS

approved

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Last modified August 7 23:50 EDT 2022. Contains 355995 sequences. (Running on oeis4.)