A353519 a(n) = 1 if n has an odd number of square divisors, otherwise 0.

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

Offset

1

Links

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

Index entries for characteristic functions.

Index entries for sequences computed from exponents in factorization of n.

Formula

Multiplicative with a(p^e) = 1 if e == {0, 1} mod 4, and 0 if e == {2, 3} mod 4.

a(n) = A000035(A046951(n)) = abs(A219009(n)).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi^2/15 (A182448). - Amiram Eldar, Jul 23 2022

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

Mathematica

f[p_, e_] := 1 + Floor[e/2]; sqdivnum[1] = 1; sqdivnum[n_] := Times @@ f @@@ FactorInteger[n]; a[n_] := If[OddQ[sqdivnum[n]], 1, 0]; Array[a, 100] (* Amiram Eldar, Jul 23 2022 *)

Prog

(PARI) A353519(n) = { my(f=factor(n)); prod(k=1, #f~, ((f[k, 2]%4)<2)); };
(PARI) A353519(n) = (sumdiv(n, d, issquare(d))%2); \\ after "isok" program in A252895

Crossrefs

Characteristic function of A252895. Parity of terms of A046951.

Absolute values of A219009.

Cf. A000035, A182448.

Sequence in context: A060038 A197774 A219009 * A267708 A167021 A267687

Adjacent sequences: A353516 A353517 A353518 * A353520 A353521 A353522

Keyword

nonn,mult

Author

Antti Karttunen, May 03 2022

Status

approved