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

Author

N. J. A. Sloane, Mar 15 1996

Keywords

• nonn
• easy
• mult

Comments

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

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

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

References

• E. 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
• Index entries for characteristic functions

Crossrefs

Cf. A000578.

Cf. A003215. - Reinhard Zumkeller, Sep 27 2008

Programs

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

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 *)

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

Formula

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(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