A076469 - OEIS

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

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OFFSET

1,2

COMMENTS

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

LINKS

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

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

MATHEMATICA

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

PROG

(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