A359592 Parity (and also absolute values) of Dirichlet inverse of A035263, where A035263(n) is parity of 2-adic valuation of 2n.

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

Offset

1

Links

Antti Karttunen, Table of n, a(n) for n = 1..100000

Index entries for characteristic functions.

Formula

Multiplicative with a(2^e) = 1 if e = 2, 0 otherwise, and for odd primes p, a(p^e) = 1 if e = 1, 0 otherwise.

a(n) = abs(A359591(n)) = A359591(n) mod 2.

a(n) = A092673(n) mod 2 = A359588(n) mod 2.

From Amiram Eldar, Jan 11 2023: (Start)

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 5/Pi^2 = 0.506605... .

Dirichlet g.f.: (zeta(s)/zeta(2*s))*(4^s+1)/(4^s+2^s). (End)

Mathematica

f[p_, e_] := If[e == 1, 1, 0]; f[2, e_] := If[e == 2, 1, 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jan 11 2023 *)

Prog

(PARI) A359592(n) = { my(f = factor(n)); prod(k=1, #f~, if(2==f[k, 1], (2==f[k, 2]), (1==f[k, 2]))); };
(Python)
from sympy import mobius
def A359592(n): return (mobius(n)&1)^(0 if n&1 else mobius(n>>1)&1) # Chai Wah Wu, Jan 09 2023

Crossrefs

Characteristic function of A091428.

Cf. A001511, A035263, A359591.

Also parity of A092673 and of A359588.

Sequence in context: A322980 A267053 A359591 * A257477 A259024 A323045

Adjacent sequences: A359589 A359590 A359591 * A359593 A359594 A359595

Keyword

nonn,mult

Author

Antti Karttunen, Jan 09 2023

Status

approved