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A382967 Biquadratefree numbers (A046100) that are not squarefree (A005117).

%I A382967 #21 Apr 22 2025 06:31:46
%S A382967 4,8,9,12,18,20,24,25,27,28,36,40,44,45,49,50,52,54,56,60,63,68,72,75,
%T A382967 76,84,88,90,92,98,99,100,104,108,116,117,120,121,124,125,126,132,135,
%U A382967 136,140,147,148,150,152,153,156,164,168,169,171,172,175,180,184
%N A382967 Biquadratefree numbers (A046100) that are not squarefree (A005117).
%C A382967 Subsequence of A252849 and first differs from it at n = 22: A252849(22) = 64 = 2^6 is not a term of this sequence.
%C A382967 Subsequence of A375229 and differs from it by not having the terms 1, 256, 512, 768, 1024, ... .
%C A382967 Numbers whose prime factorization has least one exponent that equals 2 or 3 and no higher exponent.
%C A382967 Numbers k such that 2 <= A051903(k) <= 3.
%C A382967 The asymptotic density of this sequence is 1/zeta(4) - 1/zeta(2) = A215267 - A059956 = 0.3160113... .
%H A382967 Amiram Eldar, <a href="/A382967/b382967.txt">Table of n, a(n) for n = 1..10000</a>
%F A382967 Sum_{k=1..n} a(k) ~ Pi^4 * n^2 / (12*(15 - Pi^2)). - _Vaclav Kotesovec_, Apr 11 2025
%t A382967 Select[Range[200], 2 <= Max[FactorInteger[#][[;; , 2]]] <= 3 &]
%o A382967 (PARI) isok(k) = if(k == 1, 0, my(emax = vecmax(factor(k)[, 2])); emax > 1 & emax < 4);
%o A382967 (Python)
%o A382967 from math import isqrt
%o A382967 from sympy import mobius, integer_nthroot
%o A382967 def A382967(n):
%o A382967     def bisection(f,kmin=0,kmax=1):
%o A382967         while f(kmax) > kmax: kmax <<= 1
%o A382967         kmin = kmax >> 1
%o A382967         while kmax-kmin > 1:
%o A382967             kmid = kmax+kmin>>1
%o A382967             if f(kmid) <= kmid:
%o A382967                 kmax = kmid
%o A382967             else:
%o A382967                 kmin = kmid
%o A382967         return kmax
%o A382967     def f(x): return int(n+x+sum(mobius(k)*(x//k**2-x//k**4) for k in range(1, integer_nthroot(x,4)[0]+1))+sum(mobius(k)*(x//k**2) for k in range(integer_nthroot(x,4)[0]+1,isqrt(x)+1)))
%o A382967     return bisection(f,n,n) # _Chai Wah Wu_, Apr 11 2025
%Y A382967 Disjoint union of A067259 and A375072.
%Y A382967 Intersection of A046100 and A013929.
%Y A382967 Subsequence of A252849 and A375229.
%Y A382967 Cf. A004709, A005117, A051903, A059956, A215267.
%K A382967 nonn,easy
%O A382967 1,1
%A A382967 _Amiram Eldar_, Apr 10 2025