A010057 a(n) = 1 if n is a cube, else 0.

1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset

0,1

Comments

If n has 4 divisors, a(n) = bigomega(n) - 2. - Wesley Ivan Hurt, Jun 06 2014

References

Edmund Landau, Elementary Number Theory, translation by Jacob E. Goodman of Elementare Zahlentheorie (Vol. I_1 (1927) of Vorlesungen ueber Zahlentheorie), by Edmund Landau, with added exercises by Paul T. Bateman and E. E. Kohlbecker, Chelsea Publishing Co., New York, 1958, pp. 31-32.

Links

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Peter Bala, A continuous interpolating function for A010057.

Index entries for characteristic functions.

Formula

Multiplicative with a(p^e) = 1 if 3 divides e, 0 otherwise. - Mitch Harris, Jun 09 2005

Dirichlet generating function: zeta(3s). - Franklin T. Adams-Watters, Sep 11 2005

a(n) = f(n,0) with f(x,y) = if x>0 then f(x-3*y*(y+1),y+1) else 0^(-x). - Reinhard Zumkeller, Sep 27 2008

a(A000578(n)) = 1; a(A007412(n)) = 0. - Reinhard Zumkeller, Oct 22 2011

a(n) = A000007(sum(A010872(A124010(n,k))): k = 1..A001221(n)) for n > 0. - Reinhard Zumkeller, Jun 21 2013

a(n) = 1 + floor(n^(1/3)) - ceiling(n^(1/3)). - Wesley Ivan Hurt, Jun 06 2014

a(n) = floor(n^(1/3)) - floor((n-1)^(1/3)). - Mikael Aaltonen, Feb 24 2015

From Peter Bala, Sep 12 2025: (Start)

For n >= 1, a(n) = (1/3) * lim_{x -> n} sin(Pi*x)/(x^(2/3) * sin(Pi*x^(1/3))). (Put y = x + n. If n is not a cube the limit as y -> 0 is easily seen to be 0; if n = N^3 then the limit has the indeterminate form 0/0. Apply L'Hôpital's rule.)

Similarly, for n >= 1, a(n) = (2/3) * lim_{x -> n} sin(Pi*x)/(x^(1/3)*sin(Pi*x^(2/3))).

In general, the function (1/k) * sin(Pi*x)/(x^(1-1/k) * sin(Pi*x^(1/k))), with its singularities removed, leads to a continuous function that interpolates the characteristic function of the sequence {n^k : n >= 1}. (End)

a(n) = Sum_{d|n} A210826(d) (Inverse Moebius transform of A210826). - Amiram Eldar, Dec 20 2025

Maple

A010057 := proc(n)
    if n = 0 then
        1;
    else
        for pe in ifactors(n)[2] do
            if modp(op(2, pe), 3) <> 0 then
                return 0 ;
            end if;
        end do:
    end if;
    1 ;
end proc: # R. J. Mathar, Feb 07 2023

Mathematica

Table[ Boole[ IntegerQ[n^(1/3)]], {n, 0, 80}] (* Jean-François Alcover, Jun 10 2013 *)

Prog

(Haskell)
a010057 0 = 1
a010057 n = fromEnum $ all ((== 0) . (`mod` 3)) $ a124010_row n
a010057_list = concatMap (\x -> 1 : replicate (a003215 x - 1) 0) [0..]
-- Reinhard Zumkeller, Jun 21 2013, Oct 22 2011
(PARI) a(n) = ispower(n, 3); \\ Michel Marcus, Feb 24 2015
(Python)
from sympy import integer_nthroot
def A010057(n): return int(integer_nthroot(n, 3)[1]) # Chai Wah Wu, Apr 02 2021

Crossrefs

Cf. A000578, A003215, A005369, A010052, A010054, A132918, A199918, A210826, A374016.

Sequence in context: A393068 A205988 A167700 * A359548 A359549 A359773

Adjacent sequences: A010054 A010055 A010056 * A010058 A010059 A010060

Keyword

nonn,easy,mult

Author

N. J. A. Sloane, Mar 15 1996

Status

approved