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A082293 Numbers having exactly one square divisor > 1.

%I A082293 #28 Feb 24 2025 14:37:11
%S A082293 4,8,9,12,18,20,24,25,27,28,40,44,45,49,50,52,54,56,60,63,68,75,76,84,
%T A082293 88,90,92,98,99,104,116,117,120,121,124,125,126,132,135,136,140,147,
%U A082293 148,150,152,153,156,164,168,169,171,172,175,184,188,189,198,204,207,212
%N A082293 Numbers having exactly one square divisor > 1.
%C A082293 Numbers of the form m*p^2, p prime and m squarefree (A005117). [Corrected by _Peter Munn_, Nov 17 2020]
%C A082293 The asymptotic density of this sequence is (6/Pi^2)*Sum_{n>=1} 1/prime(n)^2 = 0.274933... (A222056). - _Amiram Eldar_, Jul 07 2020
%H A082293 Amiram Eldar, <a href="/A082293/b082293.txt">Table of n, a(n) for n = 1..10000</a>
%F A082293 A046951(a(n)) = 2.
%t A082293 Select[Range[2, 200], MemberQ[{2, 3}, (e = Sort[FactorInteger[#][[;; , 2]]])[[-1]]] && (Length[e] == 1 || e[[-2]] == 1) &] (* _Amiram Eldar_, Jul 07 2020 *)
%o A082293 (PARI) is(n)=my(f=vecsort(factor(n)[,2],,4)); #f && f[1]>1 && f[1]<4 && (#f==1 || f[2]==1) \\ _Charles R Greathouse IV_, Oct 16 2015
%o A082293 (Python)
%o A082293 from math import isqrt
%o A082293 from sympy import mobius, primerange
%o A082293 def A082293(n):
%o A082293     def bisection(f,kmin=0,kmax=1):
%o A082293         while f(kmax) > kmax: kmax <<= 1
%o A082293         kmin = kmax >> 1
%o A082293         while kmax-kmin > 1:
%o A082293             kmid = kmax+kmin>>1
%o A082293             if f(kmid) <= kmid:
%o A082293                 kmax = kmid
%o A082293             else:
%o A082293                 kmin = kmid
%o A082293         return kmax
%o A082293     def g(x): return sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
%o A082293     def f(x): return int(n+x-sum(g(x//p**2) for p in primerange(isqrt(x)+1)))
%o A082293     return bisection(f,n,n) # _Chai Wah Wu_, Feb 24 2025
%Y A082293 Cf. A005117, A046951, A222056.
%Y A082293 Complement of A048111 within A013929.
%Y A082293 Subsequence of A252849.
%Y A082293 Disjoint union of A048109 and A060687.
%Y A082293 A285508 is a subsequence.
%K A082293 nonn
%O A082293 1,1
%A A082293 _Reinhard Zumkeller_, Apr 08 2003