A061704 Number of cubes dividing n.

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1

Offset

1,8

Links

Antti Karttunen, Table of n, a(n) for n = 1..10000

Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms)

Index entries for sequences computed from exponents in factorization of n

Formula

Multiplicative with a(p^e) = floor(e/3) + 1. - Mitch Harris, Apr 19 2005

G.f.: Sum_{n>=1} x^(n^3)/(1-x^(n^3)). - Joerg Arndt, Jan 30 2011

a(n) = A000005(A053150(n)).

Dirichlet g.f.: zeta(3*s)*zeta(s). - Geoffrey Critzer, Feb 07 2015

Sum_{k=1..n} a(k) ~ zeta(3)*n + zeta(1/3)*n^(1/3). - Vaclav Kotesovec, Dec 01 2020

a(n) = Sum_{k=1..n} (1 - ceiling(n/k^3) + floor(n/k^3)). - Wesley Ivan Hurt, Jan 28 2021

Example

a(128) = 3 since 128 is divisible by 1^3 = 1, 2^3 = 8 and 4^3 = 64.

Maple

N:= 1000: # to get a(1)..a(N)
G:= add(x^(n^3)/(1-x^(n^3)), n=1..floor(N^(1/3))):
S:= series(G, x, N+1):
seq(coeff(S, x, j), j=1..N); # Robert Israel, Jul 28 2017
# alternative
A061704 := proc(n)
    local a, pe ;
    a := 1 ;
    for pe in ifactors(n)[2] do
        op(2, pe) ;
        a := a*(1+floor(%/3)) ;
    end do:
    a ;
end proc:
seq(A061704(n), n=1..80) ; # R. J. Mathar, May 10 2023

Mathematica

nn = 100; f[list_, i_]:= list[[i]]; Table[ DirichletConvolve[ f[ Boole[ Map[ IntegerQ[#] &, Map[#^(1/3) &, Range[nn]]]], n], f[Table[1, {nn}], n], n, m], {m, 1, nn}] (* Geoffrey Critzer, Feb 07 2015 *)
Table[DivisorSum[n, 1 &, IntegerQ[#^(1/3)] &], {n, 105}] (* Michael De Vlieger, Jul 28 2017 *)
f[p_, e_] := 1 + Floor[e/3]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Sep 15 2020 *)

Prog

(PARI) a(n) = sumdiv(n, d, ispower(d, 3)); \\ Michel Marcus, Jan 31 2015

Crossrefs

Cf. A000005, A000578, A046951, A053150.

Sequence in context: A307428 A368248 A362852 * A325837 A366902 A050361

Adjacent sequences: A061701 A061702 A061703 * A061705 A061706 A061707

Keyword

nonn,mult

Author

Henry Bottomley, Jun 18 2001

Status

approved