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

Author

Jeppe Stig Nielsen, Mar 25 2014

Keywords

• nonn
• easy
• changed

Examples

			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.
		

Links

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

Crossrefs

Cf. A001597 (perfect powers), A097054 (nonsquare perfect powers), A340585 (noncube perfect powers).

Cf. A008683, A052409.

Programs

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

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