A353627 a(n) = 1 if the odd part of n is squarefree, otherwise 0.

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

Offset

1

Comments

Sequence gives the absolute values of A209635. See discussion there.

From Antti Karttunen, Dec 30 2022: (Start)

Note the correspondences between four sequences:

A355689 --- abs ---> A353627 (this sequence)

^ ^

| |

inv inv

| |

v v

A166486 <--- abs --- A358839

Here inv means that the sequences are Dirichlet Inverses of each other, and abs means taking absolute values.

(End)

Links

Antti Karttunen, Table of n, a(n) for n = 1..65537

Index entries for characteristic functions.

Formula

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

a(n) = A008966(A000265(n)).

a(n) = abs(A209635(n)) = 1 - A353626(n).

a(n) >= A323239(n).

a(n) >= A336477(n).

a(n) <= A353628(n).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 8/Pi^2 (A217739). - Amiram Eldar, Jul 23 2022

a(n) = abs(A355689(n)) = A359156(n) + A359158(n). - Antti Karttunen, Dec 30 2022

Dirichlet g.f.: zeta(s)/(zeta(2*s)*(1-1/4^s)). - Amiram Eldar, Jan 01 2023

Mathematica

a[n_] := If[SquareFreeQ[n/2^IntegerExponent[n, 2]], 1, 0]; Array[a, 100] (* Amiram Eldar, Jul 23 2022 *)

Prog

(PARI) A353627(n) = issquarefree(n>>valuation(n, 2));

Crossrefs

Characteristic function of A122132. Positions of zeros: A038838.

Absolute values of A209635 and A355689, binary complement of A353626.

Parity of A342417, A354347, A354823.

Sum of A359156 and A359158.

Cf. A358839 (Dirichlet inverse), A166486 (their absolute values).

Differs from A353628 at the positions given by A353456.

Cf. A000265, A008966, A217739, A336477.

Cf. also A323239, A359377.

Sequence in context: A070887 A209635 A355689 * A353628 A359551 A353458

Adjacent sequences: A353624 A353625 A353626 * A353628 A353629 A353630

Keyword

nonn,mult

Author

Antti Karttunen, May 02 2022

Status

approved