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

%I A072358 #23 Aug 13 2024 02:23:13
%S A072358 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,
%T A072358 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,
%U A072358 14,14,15,15,15,15,15,16,16,16,16,16,16,16,17,18,18,18,18,18,18,18,18,19
%N A072358 Number of cubefree numbers <= n which are not squarefree.
%H A072358 Reinhard Zumkeller, <a href="/A072358/b072358.txt">Table of n, a(n) for n = 1..10000</a>
%F A072358 a(n) = A060431(n) - A013928(n+1).
%F A072358 a(n) = Sum_{k=1..n} (A212793(k) * (1 - A008966(k))). - _Reinhard Zumkeller_, May 27 2012
%F A072358 a(n) ~ c * n, where c = 1/zeta(3) - 1/zeta(2) = A088453 - A059956 = 0.22398... - _Amiram Eldar_, Feb 16 2021
%t A072358 Accumulate @ Table[Boole[Max @ FactorInteger[n][[;; , 2]] == 2], {n, 1, 100}] (* _Amiram Eldar_, Feb 16 2021 *)
%o A072358 (Haskell)
%o A072358 a072358 n = a072358_list !! (n-1)
%o A072358 a072358_list = scanl1 (+) $
%o A072358    zipWith (*) a212793_list $ map (1 -) a008966_list
%o A072358 -- _Reinhard Zumkeller_, May 27 2012
%o A072358 (Python)
%o A072358 from math import isqrt
%o A072358 from sympy import mobius, integer_nthroot
%o A072358 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
%Y A072358 Cf. A060431, A013928.
%Y A072358 Cf. A008966, A212793.
%Y A072358 Cf. A067259, A004709, A005117, A059956, A088453.
%K A072358 nonn
%O A072358 1,9
%A A072358 _Reinhard Zumkeller_, Jul 18 2002