A060431 Number of cubefree numbers <= n.

1, 2, 3, 4, 5, 6, 7, 7, 8, 9, 10, 11, 12, 13, 14, 14, 15, 16, 17, 18, 19, 20, 21, 21, 22, 23, 23, 24, 25, 26, 27, 27, 28, 29, 30, 31, 32, 33, 34, 34, 35, 36, 37, 38, 39, 40, 41, 41, 42, 43, 44, 45, 46, 46, 47, 47, 48, 49, 50, 51, 52, 53, 54, 54, 55, 56, 57, 58, 59, 60, 61, 61

Offset: 1

Author

Vladeta Jovovic, Apr 06 2001

Keywords

• easy
• nonn

References

• I. M. Vinogradov, Elements of the Theory of Numbers,(in Russian), Moscow, 1981, p. 36.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..500 from Harry J. Smith)

Crossrefs

Cf. A013928, A013937, A088453.

Programs

Haskell

a060431 n = a060431_list !! (n-1)
a060431_list = scanl1 (+) a212793_list -- Reinhard Zumkeller, May 27 2012

Magma

[&+[MoebiusMu(d)*Floor(n div d^3):d in [1..n]]:n in [1..75]]; // Marius A. Burtea, Oct 02 2019

PARI

a(n)=sum(k=1,n,moebius(k)*floor(n/k^3)) \\ Benoit Cloitre, Jun 13 2007

PARI

for (n=1, 500, a=sum(k=1, n, moebius(k)*floor(n/k^3)); write("b060431.txt", n, " ", a)) \\ Harry J. Smith, Jul 05 2009

PARI

a(n)=my(s); forsquarefree(k=1,sqrtnint(n,3), s+=n\k[1]^3*moebius(k)); s \\ Charles R Greathouse IV, Jan 08 2018

Python

from sympy import mobius, integer_nthroot
def A060431(n): return sum(mobius(k)*(n//k**3) for k in range(1, integer_nthroot(n,3)[0]+1)) # Chai Wah Wu, Aug 06 2024

Formula

a(n) = Sum_{d>=1} mu(d)*floor(n/d^3), mu(d) = Moebius function A008683.

a(n) is asymptotic to (1/zeta(3))*n, see A088453. - Benoit Cloitre, Jun 13 2007

a(n) = Sum_{k = 1..n} A212793(k). - Reinhard Zumkeller, May 27 2012