A076468 - OEIS

%I #20 Aug 14 2024 13:11:53

%S 1,16,32,64,81,128,243,256,512,625,729,1024,1296,2048,2187,2401,3125,

%T 4096,6561,7776,8192,10000,14641,15625,16384,16807,19683,20736,28561,

%U 32768,38416,46656,50625,59049,65536,78125,83521,100000,104976,117649

%N Perfect powers m^k where k >= 4.

%C If p|n then at least p^4|n.

%C Subsequence of A036967. - _R. J. Mathar_, May 27 2011

%H Reinhard Zumkeller, <a href="/A076468/b076468.txt">Table of n, a(n) for n = 1..10000</a>

%F 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 a = {1}; Do[ If[ Apply[ GCD, Last[ Transpose[ FactorInteger[n]]]] > 3, a = Append[a, n]; Print[n]], {n, 2, 131071}]; a

%o (Haskell)

%o import qualified Data.Set as Set (null)

%o import Data.Set (empty, insert, deleteFindMin)

%o a076468 n = a076468_list !! (n-1)

%o a076468_list = 1 : f [2..] empty where

%o    f xs'@(x:xs) s | Set.null s || m > x ^ 4 = f xs $ insert (x ^ 4, x) s

%o                   | m == x ^ 4  = f xs s

%o                   | otherwise = m : f xs' (insert (m * b, b) s')

%o                   where ((m, b), s') = deleteFindMin s

%o -- _Reinhard Zumkeller_, Jun 19 2013

%o (Python)

%o from sympy import mobius, integer_nthroot

%o def A076468(n):

%o     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     kmin, kmax = 1,2

%o     while f(kmax) >= kmax:

%o         kmax <<= 1

%o     while True:

%o         kmid = kmax+kmin>>1

%o         if f(kmid) < kmid:

%o             kmax = kmid

%o         else:

%o             kmin = kmid

%o         if kmax-kmin <= 1:

%o             break

%o     return kmax # _Chai Wah Wu_, Aug 14 2024

%Y Cf. A001597, A036967, A076467, A076469, A076470.

%Y Cf. A002117, A013661.

%K nonn

%O 1,2

%A _Robert G. Wilson v_, Oct 14 2002