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

1,9

Links

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

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

Mathematica

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

Prog

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

Crossrefs

Cf. A060431, A013928.

Cf. A008966, A212793.

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

Sequence in context: A238598 A156684 A070564 * A074795 A383228 A069577

Adjacent sequences: A072355 A072356 A072357 * A072359 A072360 A072361

Keyword

nonn

Author

Reinhard Zumkeller, Jul 18 2002

Status

approved