A117453 Perfect powers in more than one way.

1, 16, 64, 81, 256, 512, 625, 729, 1024, 1296, 2401, 4096, 6561, 10000, 14641, 15625, 16384, 19683, 20736, 28561, 32768, 38416, 46656, 50625, 59049, 65536, 83521, 104976, 117649, 130321, 160000, 194481, 234256, 262144, 279841, 331776, 390625

Offset: 1

Author

Eric W. Weisstein, Mar 16 2006

Keywords

• nonn

Comments

Corresponding values of ways for a(n) in A175066(n) for n >= 2. - Jaroslav Krizek, Jan 23 2010

Perfect powers expressible as m^k with k composite. - Charlie Neder, Mar 02 2019

Examples

			16 = 2^4 = 4^2.
		

Links

• Donovan Johnson, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Perfect Power

Crossrefs

Cf. A001597, A322449.

Programs

Mathematica

s = Split@ Sort@ Flatten@ Table[ n^i, {n, 2, Sqrt@456975}, {i, 2, Log[n, 456975]}]; Union@ Flatten@ Select[s, Length@ # > 1 &] (* Robert G. Wilson v, Apr 12 2006 *)

Python

from sympy import mobius, integer_nthroot, primerange
def A117453(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return int(n+sum(mobius(k)*(integer_nthroot(x,k)[0]-1+sum(integer_nthroot(x,p*k)[0]-1 for p in primerange((x//k).bit_length()))) for k in range(1,x.bit_length())))
    return bisection(f,n,n) # Chai Wah Wu, Nov 24 2024

Extensions

More terms from Robert G. Wilson v, Apr 12 2006