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A076468 Perfect powers m^k where k >= 4.

%I A076468 #20 Aug 14 2024 13:11:53
%S A076468 1,16,32,64,81,128,243,256,512,625,729,1024,1296,2048,2187,2401,3125,
%T A076468 4096,6561,7776,8192,10000,14641,15625,16384,16807,19683,20736,28561,
%U A076468 32768,38416,46656,50625,59049,65536,78125,83521,100000,104976,117649
%N A076468 Perfect powers m^k where k >= 4.
%C A076468 If p|n then at least p^4|n.
%C A076468 Subsequence of A036967. - _R. J. Mathar_, May 27 2011
%H A076468 Reinhard Zumkeller, <a href="/A076468/b076468.txt">Table of n, a(n) for n = 1..10000</a>
%F A076468 Sum_{n>=1} 1/a(n) = 3 - zeta(2) - zeta(3) + Sum_{k>=2} mu(k)*(3 - zeta(k) - zeta(2*k) - zeta(3*k)) = 1.1473274274... . - _Amiram Eldar_, Dec 03 2022
%t A076468 a = {1}; Do[ If[ Apply[ GCD, Last[ Transpose[ FactorInteger[n]]]] > 3, a = Append[a, n]; Print[n]], {n, 2, 131071}]; a
%o A076468 (Haskell)
%o A076468 import qualified Data.Set as Set (null)
%o A076468 import Data.Set (empty, insert, deleteFindMin)
%o A076468 a076468 n = a076468_list !! (n-1)
%o A076468 a076468_list = 1 : f [2..] empty where
%o A076468    f xs'@(x:xs) s | Set.null s || m > x ^ 4 = f xs $ insert (x ^ 4, x) s
%o A076468                   | m == x ^ 4  = f xs s
%o A076468                   | otherwise = m : f xs' (insert (m * b, b) s')
%o A076468                   where ((m, b), s') = deleteFindMin s
%o A076468 -- _Reinhard Zumkeller_, Jun 19 2013
%o A076468 (Python)
%o A076468 from sympy import mobius, integer_nthroot
%o A076468 def A076468(n):
%o A076468     def f(x): return int(n+2+x-integer_nthroot(x,4)[0]-(integer_nthroot(x,6)[0]<<1)-integer_nthroot(x,9)[0]+sum(mobius(k)*(integer_nthroot(x,k)[0]+integer_nthroot(x,k<<1)[0]+integer_nthroot(x,3*k)[0]-3) for k in range(5,x.bit_length())))
%o A076468     kmin, kmax = 1,2
%o A076468     while f(kmax) >= kmax:
%o A076468         kmax <<= 1
%o A076468     while True:
%o A076468         kmid = kmax+kmin>>1
%o A076468         if f(kmid) < kmid:
%o A076468             kmax = kmid
%o A076468         else:
%o A076468             kmin = kmid
%o A076468         if kmax-kmin <= 1:
%o A076468             break
%o A076468     return kmax # _Chai Wah Wu_, Aug 14 2024
%Y A076468 Cf. A001597, A036967, A076467, A076469, A076470.
%Y A076468 Cf. A002117, A013661.
%K A076468 nonn
%O A076468 1,2
%A A076468 _Robert G. Wilson v_, Oct 14 2002