A072358 Number of cubefree numbers <= n which are not squarefree.

0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 9, 10, 10, 10, 10, 11, 12, 12, 13, 13, 13, 13, 13, 13, 13, 13, 14, 14, 14, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 16, 16, 17, 18, 18, 18, 18, 18, 18, 18, 18, 19

Offset: 1

Author

Reinhard Zumkeller, Jul 18 2002

Keywords

• nonn

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A060431, A013928.

Cf. A008966, A212793.

Cf. A067259, A004709, A005117, A059956, A088453.

Programs

Haskell

a072358 n = a072358_list !! (n-1)
a072358_list = scanl1 (+) $
   zipWith (*) a212793_list $ map (1 -) a008966_list
-- Reinhard Zumkeller, May 27 2012

Mathematica

Accumulate @ Table[Boole[Max @ FactorInteger[n][[;; , 2]] == 2], {n, 1, 100}] (* Amiram Eldar, Feb 16 2021 *)

Python

from math import isqrt
from sympy import mobius, integer_nthroot
def A072358(n): return sum(mobius(k)*(n//k**3-n//k**2) for k in range(1, integer_nthroot(n,3)[0]+1))-sum(mobius(k)*(n//k**2) for k in range(integer_nthroot(n,3)[0]+1,isqrt(n)+1)) # Chai Wah Wu, Aug 12 2024

Formula

a(n) = A060431(n) - A013928(n+1).

a(n) = Sum_{k=1..n} (A212793(k) * (1 - A008966(k))). - Reinhard Zumkeller, May 27 2012

a(n) ~ c * n, where c = 1/zeta(3) - 1/zeta(2) = A088453 - A059956 = 0.22398... - Amiram Eldar, Feb 16 2021