A239728 Perfect power but neither square nor cube.

32, 128, 243, 2048, 2187, 3125, 7776, 8192, 16807, 78125, 100000, 131072, 161051, 177147, 248832, 279936, 371293, 524288, 537824, 759375, 823543, 1419857, 1594323, 1889568, 2476099, 3200000, 4084101, 5153632, 6436343, 7962624, 8388608, 10000000, 11881376, 17210368

Offset

1,1

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Formula

GCD(A052409(a(n)), 6) = 1. - Reinhard Zumkeller, Mar 28 2014

Sum_{n>=1} 1/a(n) = 1 - zeta(2) - zeta(3) + zeta(6) + Sum_{k>=2} mu(k)*(1-zeta(k)) = 0.0448164603... - Amiram Eldar, Dec 21 2020

Example

279936 is included since 279936 = 6^7 is a power and this is not a square or a cube.
59049 = 9^5 not included since this is a square 243^2 = 59049.
32768 = 8^5 not included since this is a cube 32^3 = 32768.

Prog

(PARI) for(i=1, 2^25, if(gcd(ispower(i), 6) == 1, print(i)))
(Python)
from sympy import mobius, integer_nthroot
def A239728(n):
    def f(x): return int(n+x-integer_nthroot(x, 5)[0]+sum(mobius(k)*(integer_nthroot(x, k)[0]-1) for k in range(7, 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), A097054 (nonsquare perfect powers), A340585 (noncube perfect powers).

Cf. A008683, A052409.

Sequence in context: A271532 A264480 A247155 * A244082 A033323 A091905

Adjacent sequences: A239725 A239726 A239727 * A239729 A239730 A239731

Keyword

nonn,easy

Author

Jeppe Stig Nielsen, Mar 25 2014

Status

approved