A076468 - OEIS

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

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OFFSET

1,2

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

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

MATHEMATICA

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

PROG

(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

(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