A323239 - OEIS

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A323239

a(n) = 1 if n is odd and squarefree, otherwise a(n) = 0.

17

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

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OFFSET

0

COMMENTS

Characteristic function of A056911.

Dirichlet inverse of A166698. - Antti Karttunen, Dec 19 2022

LINKS

Antti Karttunen, Table of n, a(n) for n = 0..65537

Jon Maiga, Computer-generated formulas for A323239, Sequence Machine.

Index entries for characteristic functions.

FORMULA

a(n) = A000035(n) * A008966(n).

For n >= 1:

a(n) = abs(A087003(n)) = abs(A099991(n)).

a(n) = A085405(A156552(n)).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 4/Pi^2 (A185199). - Amiram Eldar, Jul 24 2022

a(n) = A008683(n) * A166698(n).

From Antti Karttunen, Dec 19 2022: (Start)

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

a(n) = A000035(n) - A353569(n).

(End)

Dirichlet g.f.: zeta(s)/(zeta(2*s)*(1+1/2^s)). - Amiram Eldar, Dec 27 2022

a(n) = Sum_{d|n} A359548(d). [From Sequence Machine] - Antti Karttunen, Nov 22 2023

MAPLE

f:= n -> charfcn[{true}](n::odd and numtheory:-issqrfree(n)):

map(f, [$0..200]); # Robert Israel, Jan 14 2019

MATHEMATICA

Table[If[OddQ[n]&&SquareFreeQ[n], 1, 0], {n, 0, 120}] (* Harvey P. Dale, Feb 02 2021 *)

PROG

(PARI) A323239(n) = ((n%2) && issquarefree(n));

(PARI) A323239(n) = { my(f = factor(n)); prod(k=1, #f~, (f[k, 1]%2)*(1==f[k, 2])); }; \\ Antti Karttunen, Dec 19 2022

CROSSREFS

Absolute values of A087003 and A099991.

Cf. A000035, A008683, A008966, A056911, A085405, A156552, A166698 (Dirichlet inverse), A185199, A322810, A353481, A353569, A353627.

Inverse Möbius transform of A359548.

Sequence in context: A215532 A191152 A139312 * A376451 A285685 A287773

Adjacent sequences: A323236 A323237 A323238 * A323240 A323241 A323242

KEYWORD

nonn,mult

AUTHOR

Antti Karttunen, Jan 12 2019

EXTENSIONS

Keyword:mult added by Antti Karttunen, Dec 19 2022

STATUS

approved