A340585 Noncube perfect powers.

4, 9, 16, 25, 32, 36, 49, 81, 100, 121, 128, 144, 169, 196, 225, 243, 256, 289, 324, 361, 400, 441, 484, 529, 576, 625, 676, 784, 841, 900, 961, 1024, 1089, 1156, 1225, 1296, 1369, 1444, 1521, 1600, 1681, 1764, 1849, 1936, 2025, 2048, 2116, 2187, 2209, 2304, 2401, 2500

Offset: 1

Author

Hugo Pfoertner, Jan 12 2021

Keywords

• nonn

Comments

This was the original definition of A239870. However, the true definition of that sequence seems to be slightly different.

Links

• Hugo Pfoertner, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000578, A001597, A002117, A007412, A072102, A076467, A097054, A239728.

Programs

Maple

filter:= proc(n) local g;
  g:= igcd(op(ifactors(n)[2][..,2]));
  g > 1 and (g mod 3 <> 0)
end proc:
select(filter, [$1..10000]); # Robert Israel, Jan 12 2021

Mathematica

Select[Range[2, 2500], (g = GCD @@ FactorInteger[#][[;; , 2]]) > 1 && !Divisible[g, 3] &] (* Amiram Eldar, Jan 12 2021 *)

PARI

for(n=2,2500,if( ispower(n) % 3, print1(n,", ")))

Python

from math import isqrt
from sympy import mobius, integer_nthroot
def A340585(n):
    def f(x): return int(n+x-isqrt(x)+sum(mobius(k)*(integer_nthroot(x,k)[0]-1) 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) = 1 - zeta(3) + Sum_{k>=2} mu(k)*(1-zeta(k)) = 1 - A002117 + A072102 = 0.6724074652... - Amiram Eldar, Jan 12 2021