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

1, 16, 32, 64, 81, 128, 243, 256, 512, 625, 729, 1024, 1296, 2048, 2187, 2401, 3125, 4096, 6561, 7776, 8192, 10000, 14641, 15625, 16384, 16807, 19683, 20736, 28561, 32768, 38416, 46656, 50625, 59049, 65536, 78125, 83521, 100000, 104976, 117649

Offset: 1

Author

Robert G. Wilson v, Oct 14 2002

Keywords

• nonn

Comments

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

Subsequence of A036967. - R. J. Mathar, May 27 2011

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Crossrefs

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

Cf. A002117, A013661.

Programs

Haskell

import qualified Data.Set as Set (null)
import Data.Set (empty, insert, deleteFindMin)
a076468 n = a076468_list !! (n-1)
a076468_list = 1 : f [2..] empty where
   f xs'@(x:xs) s | Set.null s || m > x ^ 4 = f xs $ insert (x ^ 4, x) s
                  | m == x ^ 4  = f xs s
                  | otherwise = m : f xs' (insert (m * b, b) s')
                  where ((m, b), s') = deleteFindMin s
-- Reinhard Zumkeller, Jun 19 2013

Mathematica

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

Python

from sympy import mobius, integer_nthroot
def A076468(n):
    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())))
    kmin, kmax = 1,2
    while f(kmax) >= kmax:
        kmax <<= 1
    while True:
        kmid = kmax+kmin>>1
        if f(kmid) < kmid:
            kmax = kmid
        else:
            kmin = kmid
        if kmax-kmin <= 1:
            break
    return kmax # Chai Wah Wu, Aug 14 2024

Formula

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