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

1,1

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

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

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A331219 A337534 A126589 * A010409 A010457 A244833

Adjacent sequences: A340582 A340583 A340584 * A340586 A340587 A340588

Keyword

nonn

Author

Hugo Pfoertner, Jan 12 2021

Status

approved