A076469 Perfect powers m^k where k >= 5.

1, 32, 64, 128, 243, 256, 512, 729, 1024, 2048, 2187, 3125, 4096, 6561, 7776, 8192, 15625, 16384, 16807, 19683, 32768, 46656, 59049, 65536, 78125, 100000, 117649, 131072, 161051, 177147, 248832, 262144, 279936, 371293, 390625, 524288, 531441

Offset: 1

Author

Robert G. Wilson v, Oct 14 2002

Keywords

• nonn

Comments

If p|n when at least p^5|n.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A001597, A076467, A076468, A076470.

Cf. A002117, A013661, A013662.

Programs

Mathematica

a = {1}; Do[ If[ Apply[ GCD, Last[ Transpose[ FactorInteger[n]]]] > 4, a = Append[a, n]; Print[n]], {n, 2, 537823}]; a

Python

from sympy import mobius, integer_nthroot
def A076469(n):
    def f(x): return int(n+3+x-(integer_nthroot(x,6)[0]<<1)-integer_nthroot(x,8)[0]-integer_nthroot(x,9)[0]-integer_nthroot(x,12)[0]+sum(mobius(k)*(integer_nthroot(x,k)[0]+integer_nthroot(x,k<<1)[0]+integer_nthroot(x,3*k)[0]+integer_nthroot(x,k<<2)[0]-4) 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

Formula

Sum_{n>=1} 1/a(n) = 4 - zeta(2) - zeta(3) - zeta(4) + Sum_{k>=2} mu(k)*(4 - zeta(k) - zeta(2*k) - zeta(3*k) - zeta(4*k)) = 1.06932853458... . - Amiram Eldar, Dec 03 2022