A097054 Nonsquare perfect powers.

8, 27, 32, 125, 128, 216, 243, 343, 512, 1000, 1331, 1728, 2048, 2187, 2197, 2744, 3125, 3375, 4913, 5832, 6859, 7776, 8000, 8192, 9261, 10648, 12167, 13824, 16807, 17576, 19683, 21952, 24389, 27000, 29791, 32768, 35937, 39304, 42875, 50653

Offset

1,1

Comments

Terms of A001597 that are not in A000290.

All terms of this sequence are also in A070265 (odd powers), but omitting those odd powers that are also a square (e.g. 64=4^3=8^2).

Links

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

Eric Weisstein's World of Mathematics, Perfect Power.

Eric Weisstein's World of Mathematics, Odd Power.

Formula

A052409(a(n)) is odd. - Reinhard Zumkeller, Mar 28 2014

Sum_{n>=1} 1/a(n) = 1 - zeta(2) + Sum_{k>=2} mu(k)*(1-zeta(k)) = 0.2295303015... - Amiram Eldar, Dec 21 2020

Maple

# uses code of A001597
for n from 4 do
    if not issqr(n) and isA001597(n) then
        printf("%d, \n", n);
    end if;
end do: # R. J. Mathar, Jan 13 2021

Mathematica

nn = 50653; Select[Union[Flatten[Table[n^i, {i, Prime[Range[2, PrimePi[Log[2, nn]]]]}, {n, 2, nn^(1/i)}]]], ! IntegerQ[Sqrt[#]] &] (* T. D. Noe, Apr 19 2011 *)

Prog

(Haskell)
import Data.Map (singleton, findMin, deleteMin, insert)
a097054 n = a097054_list !! (n-1)
a097054_list = f 9 (3, 2) (singleton 4 (2, 2)) where
   f zz (bz, be) m
    | xx < zz && even be =
                f zz (bz, be+1) (insert (bx*xx) (bx, be+1) $ deleteMin m)
    | xx < zz = xx :
                f zz (bz, be+1) (insert (bx*xx) (bx, be+1) $ deleteMin m)
    | xx > zz = f (zz+2*bz+1) (bz+1, 2) (insert (bz*zz) (bz, 3) m)
    | otherwise = f (zz + 2 * bz + 1) (bz + 1, 2) m
    where (xx, (bx, be)) = findMin m
-- Reinhard Zumkeller, Mar 28 2014
(PARI) is(n)=ispower(n)%2 \\ Charles R Greathouse IV, Aug 28 2016
(PARI) list(lim)=my(v=List()); forprime(e=3, logint(lim\=1, 2), for(b=2, sqrtnint(lim, e), if(!issquare(b), listput(v, b^e)))); Set(v) \\ Charles R Greathouse IV, Jan 09 2023
(Python)
from sympy import mobius, integer_nthroot
def A097054(n):
    def f(x): return int(n-1+x+sum(mobius(k)*(integer_nthroot(x, k)[0]-1) for k in range(3, 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

Crossrefs

Cf. A001597 (perfect powers), A000290 (the squares), A008683, A070265 (odd powers), A097055, A097056, A239870, A239728, A093771.

Sequence in context: A339595 A376173 A335988 * A370788 A304291 A056824

Adjacent sequences: A097051 A097052 A097053 * A097055 A097056 A097057

Keyword

nonn,easy

Author

Hugo Pfoertner, Jul 21 2004

Status

approved