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A384518 Nonsquarefree numbers that are squarefree numbers raised to an odd power.

%I A384518 #15 Jun 01 2025 16:37:57
%S A384518 8,27,32,125,128,216,243,343,512,1000,1331,2048,2187,2197,2744,3125,
%T A384518 3375,4913,6859,7776,8192,9261,10648,12167,16807,17576,19683,24389,
%U A384518 27000,29791,32768,35937,39304,42875,50653,54872,59319,68921,74088,78125,79507,97336,100000
%N A384518 Nonsquarefree numbers that are squarefree numbers raised to an odd power.
%C A384518 Subsequence of A097054 and first differs from it at n = 12: A097054(12) = 1728 = 2^6 * 3^3 is not a term of this sequence.
%C A384518 Numbers whose prime factorization exponents are equal, odd and larger than 1.
%H A384518 Amiram Eldar, <a href="/A384518/b384518.txt">Table of n, a(n) for n = 1..10000</a>
%H A384518 <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>.
%H A384518 <a href="/index/Pow#powerful">Index entries for sequences related to powerful numbers</a>.
%F A384518 Sum_{n>=1} 1/a(n) = Sum_{k>=1} (zeta(2*k+1)/zeta(4*k+2)-1) = 0.22841193284408713846... .
%t A384518 Select[Range[10^5], Length[(u = Union[FactorInteger[#][[;; , 2]]])] == 1 && u[[1]] > 1 && OddQ[u[[1]]] &]
%o A384518 (PARI) isok(k) = {my(s, e = ispower(k, , &s)); e % 2 && issquarefree(s);}
%o A384518 (Python)
%o A384518 from math import isqrt
%o A384518 from sympy import mobius, integer_nthroot
%o A384518 def A384518(n):
%o A384518     def bisection(f,kmin=0,kmax=1):
%o A384518         while f(kmax) > kmax: kmax <<= 1
%o A384518         kmin = kmax >> 1		
%o A384518         while kmax-kmin > 1:
%o A384518             kmid = kmax+kmin>>1
%o A384518             if f(kmid) <= kmid:
%o A384518                 kmax = kmid
%o A384518             else:
%o A384518                 kmin = kmid
%o A384518         return kmax
%o A384518     def g(x): return sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
%o A384518     def f(x): return n+x-sum(g(integer_nthroot(x,e)[0])-1 for e in range(3,x.bit_length(),2))
%o A384518     return bisection(f,n,n) # _Chai Wah Wu_, Jun 01 2025
%Y A384518 Intersection of A072777 and A268335.
%Y A384518 Equals A072777 \ A384517.
%Y A384518 Subsequence of A097054.
%Y A384518 Cf. A005117.
%K A384518 nonn
%O A384518 1,1
%A A384518 _Amiram Eldar_, Jun 01 2025