A376139 a(n) = (-1)^A051903(n).

1, -1, -1, 1, -1, -1, -1, -1, 1, -1, -1, 1, -1, -1, -1, 1, -1, 1, -1, 1, -1, -1, -1, -1, 1, -1, -1, 1, -1, -1, -1, -1, -1, -1, -1, 1, -1, -1, -1, -1, -1, -1, -1, 1, 1, -1, -1, 1, 1, 1, -1, 1, -1, -1, -1, -1, -1, -1, -1, 1, -1, -1, 1, 1, -1, -1, -1, 1, -1, -1, -1

Offset

1

Comments

The asymptotic density of the occurrences of 1's is Sum_{k>=2} (-1)^k * (1 - 1/zeta(k)) = 0.27591672059822700769..., which is the asymptotic density of A368714.

Links

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

Index entries for sequences computed from exponents in factorization of n.

Formula

a(n) = 1 if and only if n is in A368714.

a(n) = -1 if n is a squarefree number (A005117) that is larger than 1.

Asymptotic mean: c = Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2 * (Sum_{k>=2} (-1)^k * (1 - 1/zeta(k))) - 1 = -0.44816655880354598461... . The asymptotic standard deviation of this sequence is sqrt(1-c^2) = 0.89395007442820192379... .

Mathematica

a[n_] := (-1)^If[n == 1, 0, Max[FactorInteger[n][[;; , 2]]]]; Array[a, 100]

Prog

(PARI) a(n) = (-1)^if(n == 1, 0, vecmax(factor(n)[, 2]));
(Python)
from sympy import factorint
def A376139(n): return -1 if max(factorint(n).values(), default=0)&1 else 1 # Chai Wah Wu, Sep 12 2024

Crossrefs

Cf. A005117, A051903, A368714.

Sequence in context: A160357 A186039 A332433 * A374385 A057077 A262725

Adjacent sequences: A376136 A376137 A376138 * A376140 A376141 A376142

Keyword

sign,easy

Author

Amiram Eldar, Sep 11 2024

Status

approved