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A358040 a(n) is the number of divisors of the n-th cubefree number.

%I A358040 #22 Aug 06 2024 02:10:28
%S A358040 1,2,2,3,2,4,2,3,4,2,6,2,4,4,2,6,2,6,4,4,2,3,4,6,2,8,2,4,4,4,9,2,4,4,
%T A358040 2,8,2,6,6,4,2,3,6,4,6,2,4,4,4,2,12,2,4,6,4,8,2,6,4,8,2,2,4,6,6,4,8,2,
%U A358040 4,2,12,4,4,4,2,12,4,6,4,4,4,2,6,6,9,2
%N A358040 a(n) is the number of divisors of the n-th cubefree number.
%C A358040 The analogous sequence with squarefree numbers is A072048.
%H A358040 Amiram Eldar, <a href="/A358040/b358040.txt">Table of n, a(n) for n = 1..10000</a>
%H A358040 Zhu Weiyi, <a href="https://fs.unm.edu/SNJ/OnTheCubeFreeNumberSequence.pdf">On the cube free number sequences</a>, Smarandache Notions J., Vol. 14 (2004), pp. 199-202.
%F A358040 a(n) = A000005(A004709(n)).
%F A358040 Sum_{k=1..n} a(k) = (36*c_1/Pi^4) * n * (log(n) + (2*gamma - 1) - 24*zeta'(2)/Pi^2 - 4*c_2) + O(n^(1/2 + eps)), where c_1 = Product_{p prime} ((p^2+2*p+3)/(p+1)^2) = 1.58095136661854869148023... and c_2 = Sum_{p prime} p*log(p)/((p+1)*(p^2+2*p+3)) = 0.229224... (Weiyi, 2004).
%t A358040 DivisorSigma[0, Select[Range[100], Max[FactorInteger[#][[;;, 2]]] < 3 &]]
%o A358040 (Python)
%o A358040 from sympy import mobius, integer_nthroot, divisor_count
%o A358040 def A358040(n):
%o A358040     def f(x): return n+x-sum(mobius(k)*(x//k**3) for k in range(1, integer_nthroot(x,3)[0]+1))
%o A358040     m, k = n, f(n)
%o A358040     while m != k:
%o A358040         m, k = k, f(k)
%o A358040     return divisor_count(m) # _Chai Wah Wu_, Aug 06 2024
%Y A358040 Cf. A000005, A001620 (gamma), A004709, A072048, A073002 (-zeta'(2)), A147533 (2*gamma-1), A358039.
%K A358040 nonn
%O A358040 1,2
%A A358040 _Amiram Eldar_, Oct 29 2022