A060431 - OEIS

%I #34 Aug 06 2024 02:12:32

%S 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,

%T 23,24,25,26,27,27,28,29,30,31,32,33,34,34,35,36,37,38,39,40,41,41,42,

%U 43,44,45,46,46,47,47,48,49,50,51,52,53,54,54,55,56,57,58,59,60,61,61

%N Number of cubefree numbers <= n.

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

%H Amiram Eldar, <a href="/A060431/b060431.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..500 from Harry J. Smith)

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

%F a(n) is asymptotic to (1/zeta(3))*n, see A088453. - _Benoit Cloitre_, Jun 13 2007

%F a(n) = Sum_{k = 1..n} A212793(k). - _Reinhard Zumkeller_, May 27 2012

%o (PARI) a(n)=sum(k=1,n,moebius(k)*floor(n/k^3)) \\ _Benoit Cloitre_, Jun 13 2007

%o (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

%o (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

%o (Haskell)

%o a060431 n = a060431_list !! (n-1)

%o a060431_list = scanl1 (+) a212793_list -- _Reinhard Zumkeller_, May 27 2012

%o (Magma) [&+[MoebiusMu(d)*Floor(n div d^3):d in [1..n]]:n in [1..75]]; // _Marius A. Burtea_, Oct 02 2019

%o (Python)

%o from sympy import mobius, integer_nthroot

%o 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

%Y Cf. A013928, A013937, A088453.

%K easy,nonn

%O 1,2

%A _Vladeta Jovovic_, Apr 06 2001