A008834 Largest cube dividing n.

1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 27, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 27, 1, 8, 1, 1, 1, 1, 1, 1, 1, 64, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 8, 27

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• mult

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000
• Henry Bottomley, Some Smarandache-type multiplicative sequences.
• Vaclav Kotesovec, Graph - the asymptotic ratio.
• Eric Weisstein's World of Mathematics, Cubic Part.

Crossrefs

Cf. A000203, A050985, A053150.

Programs

Maple

with(numtheory): [ seq( expand(nthpow(i,3)),i=1..200) ];
# alternative:
A008834 := proc(n)
    local p;
    a := 1 ;
    for p in ifactors(n)[2] do
        e := floor(op(2,p)/3) ;
        a := a*op(1,p)^(3*e) ;
    end do:
    a ;
end proc:
seq(A008834(n),n=1..40) ; # R. J. Mathar, Dec 08 2015

Mathematica

a[n_] := Times @@ (#[[1]]^(#[[2]] - Mod[#[[2]], 3]) & ) /@ FactorInteger[n]; Table[a[n], {n, 1, 81}]
(* Jean-François Alcover, Jul 31 2011, after PARI prog. *)
  upto=1000;Flatten[With[{c=Range[Floor[Surd[upto,3]],1,-1]^3}, Table[ Select[ c,Divisible[n,#]&,1],{n,upto}]]](* Harvey P. Dale, Apr 07 2013 *)

PARI

a(n)=n=factor(n);prod(i=1,#n[,1],n[i,1]^(n[i,2]\3*3)) \\ Charles R Greathouse IV, Jul 28 2011

Python

from math import prod
from sympy import factorint
def A008834(n): return prod(p**(e-e%3) for p, e in factorint(n).items()) # Chai Wah Wu, Aug 08 2024

Formula

Multiplicative with a(p^e) = p^(3[e/3]). - Mitch Harris, Apr 19 2005

a(n) = A053150(n)^3. - R. J. Mathar, May 27 2011

Dirichlet g.f.: zeta(s)*zeta(3s-3)/zeta(3s). The Dirichlet convolution of this sequence with A050985 generates A000203. - R. J. Mathar, Apr 05 2011

Sum_{k=1..n} a(k) ~ 45 * zeta(4/3) * n^(4/3) / (2*Pi^4). - Vaclav Kotesovec, Jan 31 2019

a(n) = n/A050985(n). - Amiram Eldar, Aug 15 2023