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A359377
a(n) = 1 if 3*n is squarefree, otherwise 0.
9
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.
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
KEYWORD
nonn,mult
AUTHOR
Antti Karttunen, Dec 29 2022
STATUS
approved