A075090 Even perfect powers: numbers of the form (2*m)^k for some m>=1 and k>=2.

4, 8, 16, 32, 36, 64, 100, 128, 144, 196, 216, 256, 324, 400, 484, 512, 576, 676, 784, 900, 1000, 1024, 1156, 1296, 1444, 1600, 1728, 1764, 1936, 2048, 2116, 2304, 2500, 2704, 2744, 2916, 3136, 3364, 3600, 3844, 4096, 4356, 4624, 4900, 5184, 5476, 5776, 5832

Offset

1,1

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Chai Wah Wu, Algorithms for Complementary Sequences, Integers (2025) Vol. 25, Art. No. A95. See p. 24.

Formula

Sum_{n>=1} 1/a(n) = -Sum_{k>=2} mu(k)*zeta(k)/2^k = 0.5854268109... - Amiram Eldar, Dec 19 2020

Maple

q:= n-> n::even and igcd(ifactors(n)[2][.., 2][])>1:
select(q, [$1..6000])[];  # Alois P. Heinz, May 04 2022

Mathematica

Take[Union[Flatten[Table[a^b, {a, 2, 100, 2}, {b, 2, 15}]]], 50] (* Alonso del Arte, Nov 22 2011 *)

Prog

(Haskell)
a075090 n = a075090_list !! (n-1)
a075090_list = filter even a001597_list  -- Reinhard Zumkeller, Oct 04 2012
(PARI) isok(m) = !(m%2) && ispower(m); \\ Michel Marcus, May 03 2022
(Python)
from sympy import mobius, integer_nthroot
from oeis_sequences.OEISsequences import bisection
def A075090(n):
    def f(x): return int(n+x+sum(mobius(k)*(integer_nthroot(x, k)[0]>>1) for k in range(2, x.bit_length())))
    return bisection(f, n, n) # Chai Wah Wu, Sep 15 2025

Crossrefs

Intersection of A005843 and A001597.

Cf. A008683, A075109.

Sequence in context: A387901 A181800 A319180 * A354955 A304250 A368681

Adjacent sequences: A075087 A075088 A075089 * A075091 A075092 A075093

Keyword

easy,nonn

Author

Zak Seidov, Oct 11 2002

Extensions

Name formula corrected by Kevin Ryde, May 04 2022

Status

approved