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A375072 Biquadratefree numbers (A046100) that are not cubefree (A004709).

%I A375072 #14 Aug 06 2024 02:13:34
%S A375072 8,24,27,40,54,56,72,88,104,108,120,125,135,136,152,168,184,189,200,
%T A375072 216,232,248,250,264,270,280,296,297,312,328,343,344,351,360,375,376,
%U A375072 378,392,408,424,440,456,459,472,488,500,504,513,520,536,540,552,568,584,594,600
%N A375072 Biquadratefree numbers (A046100) that are not cubefree (A004709).
%C A375072 Subsequence of A176297 and first differs from it at n = 41: A176297(41) = 432 = 2^4 * 3^3 is not a term of this sequence.
%C A375072 Numbers whose prime factorization has least one exponent that equals 3 and no higher exponent.
%C A375072 Numbers k such that A051903(k) = 3.
%C A375072 The asymptotic density of this sequence is 1/zeta(4) - 1/zeta(3) = A215267 - A088453 = 0.0920310303408826983406... .
%H A375072 Amiram Eldar, <a href="/A375072/b375072.txt">Table of n, a(n) for n = 1..10000</a>
%H A375072 <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>.
%t A375072 Select[Range[600], Max[FactorInteger[#][[;; , 2]]] == 3 &]
%o A375072 (PARI) is(k) = k > 1 && vecmax(factor(k)[,2]) == 3;
%o A375072 (Python)
%o A375072 from sympy import mobius, integer_nthroot
%o A375072 def A375072(n):
%o A375072     def f(x): return n+x-sum(mobius(k)*(x//k**4-x//k**3) for k in range(1, integer_nthroot(x,4)[0]+1))+sum(mobius(k)*(x//k**3) for k in range(integer_nthroot(x,4)[0]+1, integer_nthroot(x,3)[0]+1))
%o A375072     m, k = n, f(n)
%o A375072     while m != k:
%o A375072         m, k = k, f(k)
%o A375072     return m # _Chai Wah Wu_, Aug 05 2024
%Y A375072 Intersection of A046100 and A176297.
%Y A375072 Cf. A004709, A051903, A067259.
%Y A375072 Cf. A002117, A013662, A088453, A215267.
%K A375072 nonn,easy
%O A375072 1,1
%A A375072 _Amiram Eldar_, Jul 29 2024