A375349 a(n) is the parity of the n-th cubefree number.

1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0

Offset

1

Links

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

Formula

a(n) = A004709(n) mod 2 = A000035(A004709(n)).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 4/7.

In general, the asymptotic mean of the parity of the k-free numbers is 2^(k-1)/(2^k - 1).

Mathematica

cubeFreeQ[n_] := Max[FactorInteger[n][[;; , 2]]] < 3; Mod[Select[Range[200], cubeFreeQ], 2]

Prog

(PARI) lista(kmax) = print1(1, ", "); for(k = 2, kmax, if(vecmax(factor(k)[, 2]) < 3, print1(k % 2, ", ")));
(Python)
from sympy import mobius, integer_nthroot
def A375349(n):
    def f(x): return n+x-sum(mobius(k)*(x//k**3) for k in range(1, integer_nthroot(x, 3)[0]+1))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return m&1 # Chai Wah Wu, Aug 13 2024

Crossrefs

Cf. A000035, A004709, A373549, A373550.

Sequence in context: A191188 A285592 A276793 * A284364 A115788 A195062

Adjacent sequences: A375346 A375347 A375348 * A375350 A375351 A375352

Keyword

nonn,easy

Author

Amiram Eldar, Aug 12 2024

Status

approved