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A366536 The number of unitary divisors of the cubefree numbers (A004709).

%I A366536 #17 Aug 05 2024 17:35:50
%S A366536 1,2,2,2,2,4,2,2,4,2,4,2,4,4,2,4,2,4,4,4,2,2,4,4,2,8,2,4,4,4,4,2,4,4,
%T A366536 2,8,2,4,4,4,2,2,4,4,4,2,4,4,4,2,8,2,4,4,4,8,2,4,4,8,2,2,4,4,4,4,8,2,
%U A366536 4,2,8,4,4,4,2,8,4,4,4,4,4,2,4,4,4,2,8
%N A366536 The number of unitary divisors of the cubefree numbers (A004709).
%C A366536 The number of unitary divisors of the squarefree numbers (A005117) is the same as the number of divisors of the squarefree numbers (A072048), because all the divisors of a squarefree number are unitary.
%H A366536 Amiram Eldar, <a href="/A366536/b366536.txt">Table of n, a(n) for n = 1..10000</a>
%F A366536 a(n) = A034444(A004709(n)).
%t A366536 f[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, If[AllTrue[e, # < 3 &], 2^Length[e], Nothing]]; f[1] = 1; Array[f, 150]
%o A366536 (PARI) lista(max) = for(k = 1, max, my(e = factor(k)[, 2], iscubefree = 1); for(i = 1, #e, if(e[i] > 2, iscubefree = 0; break)); if(iscubefree, print1(2^(#e), ", ")));
%o A366536 (Python)
%o A366536 from sympy.ntheory.factor_ import udivisor_count
%o A366536 from sympy import mobius, integer_nthroot
%o A366536 def A366536(n):
%o A366536     def f(x): return n+x-sum(mobius(k)*(x//k**3) for k in range(1, integer_nthroot(x,3)[0]+1))
%o A366536     m, k = n, f(n)
%o A366536     while m != k:
%o A366536         m, k = k, f(k)
%o A366536     return udivisor_count(m) # _Chai Wah Wu_, Aug 05 2024
%Y A366536 Cf. A004709, A005117, A034444, A072048, A077610, A358040, A366440, A366537.
%Y A366536 Similar sequences: A366534, A366538.
%K A366536 nonn,easy
%O A366536 1,2
%A A366536 _Amiram Eldar_, Oct 12 2023