A359377 a(n) = 1 if 3*n is squarefree, otherwise 0.

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

Offset

1

Comments

Note the correspondences between four sequences:

A156277 --- abs ---> A359377 (this sequence)

^ ^

| |

inv inv

| |

v v

A011655 <--- abs --- A359378

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

Links

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

Index entries for characteristic functions.

Formula

Multiplicative with a(3^e) = 0, and for primes p <> 3, a(p^e) = 1 if e = 1, and 0 if e > 1.

a(n) = A008966(3*n).

a(n) = abs(A156277(n)).

a(n) = A000035(A349125(n)).

From Amiram Eldar, Jan 03 2023: (Start)

Dirichlet g.f.: zeta(s)*(1-1/3^s).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 9/(2*Pi^2) = 0.455945... (A088245). (End)

a(n) = A011655(|A055615(n)|) = abs(A365428(n)). - Antti Karttunen, Sep 16 2023

Mathematica

a[n_] := If[SquareFreeQ[3*n], 1, 0]; Array[a, 100] (* Amiram Eldar, Dec 30 2022 *)

Prog

(PARI) A359377(n) = issquarefree(3*n);
(PARI) A359377(n) = { my(f = factor(n)); prod(k=1, #f~, ((3!=f[k, 1])&&(1==f[k, 2]))); };

Crossrefs

Characteristic function of A261034.

Absolute values of A156277 and of A365428.

Cf. A000035, A008966, A088245, A011655, A055615, A349125, A359378 (Dirichlet inverse).

Cf. also A323239, A353627.

Sequence in context: A319448 A365428 A156277 * A353663 A260595 A328102

Adjacent sequences: A359374 A359375 A359376 * A359378 A359379 A359380

Keyword

nonn,mult

Author

Antti Karttunen, Dec 29 2022

Status

approved