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A076470 Perfect powers m^k where k >= 6.

%I A076470 #17 Aug 14 2024 13:11:58
%S A076470 1,64,128,256,512,729,1024,2048,2187,4096,6561,8192,15625,16384,19683,
%T A076470 32768,46656,59049,65536,78125,117649,131072,177147,262144,279936,
%U A076470 390625,524288,531441,823543,1000000,1048576,1594323,1679616,1771561
%N A076470 Perfect powers m^k where k >= 6.
%C A076470 A necessary but not sufficient condition is that if p|n when at least p^6|n.
%H A076470 Amiram Eldar, <a href="/A076470/b076470.txt">Table of n, a(n) for n = 1..10000</a>
%F A076470 Sum_{n>=1} 1/a(n) = 5 - zeta(2) - zeta(3) - zeta(4) - zeta(5) + Sum_{k>=2} mu(k)*(5 - zeta(k) - zeta(2*k) - zeta(3*k) - zeta(4*k) - zeta(5*k)) = 1.03342597171... . - _Amiram Eldar_, Dec 03 2022
%t A076470 a = {1}; Do[ If[ Apply[ GCD, Last[ Transpose[ FactorInteger[n]]]] > 4, a = Append[a, n]; Print[n]], {n, 2, 1953124}]; a
%o A076470 (Python)
%o A076470 from sympy import mobius, integer_nthroot
%o A076470 def A076470(n):
%o A076470     def f(x): return int(n+9+x-(sum(integer_nthroot(x,d)[0] for d in (6,10,15))<<1)-sum(integer_nthroot(x,d)[0] for d in (8,9,12,20,25))+sum(mobius(k)*(sum(integer_nthroot(x,k*i)[0] for i in range(1,6))-5) for k in range(6,x.bit_length())))
%o A076470     kmin, kmax = 1,2
%o A076470     while f(kmax) >= kmax:
%o A076470         kmax <<= 1
%o A076470     while True:
%o A076470         kmid = kmax+kmin>>1
%o A076470         if f(kmid) < kmid:
%o A076470             kmax = kmid
%o A076470         else:
%o A076470             kmin = kmid
%o A076470         if kmax-kmin <= 1:
%o A076470             break
%o A076470     return kmax # _Chai Wah Wu_, Aug 14 2024
%Y A076470 Cf. A001597, A076467, A076468, A076469.
%Y A076470 Different from A069493.
%Y A076470 Cf. A002117, A013661, A013662, A013663.
%K A076470 nonn
%O A076470 1,2
%A A076470 _Robert G. Wilson v_, Oct 14 2002