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a(n) is the number of divisors of the n-th cubefree number.
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%I #22 Aug 06 2024 02:10:28

%S 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 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 4,2,12,4,4,4,2,12,4,6,4,4,4,2,6,6,9,2

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

%C The analogous sequence with squarefree numbers is A072048.

%H Amiram Eldar, <a href="/A358040/b358040.txt">Table of n, a(n) for n = 1..10000</a>

%H 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 a(n) = A000005(A004709(n)).

%F 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 DivisorSigma[0, Select[Range[100], Max[FactorInteger[#][[;;, 2]]] < 3 &]]

%o (Python)

%o from sympy import mobius, integer_nthroot, divisor_count

%o def A358040(n):

%o def f(x): return n+x-sum(mobius(k)*(x//k**3) for k in range(1, integer_nthroot(x,3)[0]+1))

%o m, k = n, f(n)

%o while m != k:

%o m, k = k, f(k)

%o return divisor_count(m) # _Chai Wah Wu_, Aug 06 2024

%Y Cf. A000005, A001620 (gamma), A004709, A072048, A073002 (-zeta'(2)), A147533 (2*gamma-1), A358039.

%K nonn

%O 1,2

%A _Amiram Eldar_, Oct 29 2022