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A076469 Perfect powers m^k where k >= 5.

%I A076469 #15 Aug 14 2024 13:11:48
%S A076469 1,32,64,128,243,256,512,729,1024,2048,2187,3125,4096,6561,7776,8192,
%T A076469 15625,16384,16807,19683,32768,46656,59049,65536,78125,100000,117649,
%U A076469 131072,161051,177147,248832,262144,279936,371293,390625,524288,531441
%N A076469 Perfect powers m^k where k >= 5.
%C A076469 If p|n when at least p^5|n.
%H A076469 Amiram Eldar, <a href="/A076469/b076469.txt">Table of n, a(n) for n = 1..10000</a>
%F A076469 Sum_{n>=1} 1/a(n) = 4 - zeta(2) - zeta(3) - zeta(4) + Sum_{k>=2} mu(k)*(4 - zeta(k) - zeta(2*k) - zeta(3*k) - zeta(4*k)) = 1.06932853458... . - _Amiram Eldar_, Dec 03 2022
%t A076469 a = {1}; Do[ If[ Apply[ GCD, Last[ Transpose[ FactorInteger[n]]]] > 4, a = Append[a, n]; Print[n]], {n, 2, 537823}]; a
%o A076469 (Python)
%o A076469 from sympy import mobius, integer_nthroot
%o A076469 def A076469(n):
%o A076469     def f(x): return int(n+3+x-(integer_nthroot(x,6)[0]<<1)-integer_nthroot(x,8)[0]-integer_nthroot(x,9)[0]-integer_nthroot(x,12)[0]+sum(mobius(k)*(integer_nthroot(x,k)[0]+integer_nthroot(x,k<<1)[0]+integer_nthroot(x,3*k)[0]+integer_nthroot(x,k<<2)[0]-4) for k in range(5,x.bit_length())))
%o A076469     kmin, kmax = 1,2
%o A076469     while f(kmax) >= kmax:
%o A076469         kmax <<= 1
%o A076469     while True:
%o A076469         kmid = kmax+kmin>>1
%o A076469         if f(kmid) < kmid:
%o A076469             kmax = kmid
%o A076469         else:
%o A076469             kmin = kmid
%o A076469         if kmax-kmin <= 1:
%o A076469             break
%o A076469     return kmax # _Chai Wah Wu_, Aug 14 2024
%Y A076469 Cf. A001597, A076467, A076468, A076470.
%Y A076469 Cf. A002117, A013661, A013662.
%K A076469 nonn
%O A076469 1,2
%A A076469 _Robert G. Wilson v_, Oct 14 2002