A373154 - OEIS

%I #12 May 29 2024 07:05:34

%S 1,0,0,0,1,0,1,0,0,0,1,0,1,0,0,0,1,0,1,0,0,0,1,0,0,0,0,0,1,0,1,0,0,0,

%T 1,0,1,0,0,0,1,0,1,0,0,0,1,0,0,0,0,0,1,0,1,0,0,0,1,0,1,0,0,0,1,0,1,0,

%U 0,0,1,0,1,0,0,0,1,0,1,0,0,0,1,0,1,0,0,0,1,0,1,0,0,0,1,0,1,0,0,0,1,0,1,0,0,0,1,0,1,0,0,0,1,0,1,0,0,0,1,0,0

%N a(n) = 1 if 6*n is squarefree, otherwise 0.

%H Antti Karttunen, <a href="/A373154/b373154.txt">Table of n, a(n) for n = 1..100000</a>

%H <a href="/index/Ch#char_fns">Index entries for characteristic functions</a>.

%F Multiplicative with a(p^e) = 0 if p <= 3 or e >= 2, and 1 otherwise.

%F a(n) = A008966(A008588(n)) = A008966(n) * A354354(n).

%F a(n) = abs(A355688(n)) = A355688(n) mod 2.

%F From _Amiram Eldar_, May 29 2024: (Start)

%F Dirichlet g.f.: (2^s/(2^s+1)) * (3^s/(3^s+1)) * zeta(s)/zeta(2*s).

%F Sum_{k=1..n} a(k) ~ (3/Pi^2) * n. (End)

%t a[n_] := If[SquareFreeQ[6*n], 1, 0]; Array[a, 100] (* _Amiram Eldar_, May 29 2024 *)

%o (PARI) A373154(n) = issquarefree(6*n);

%o (PARI) A373154(n) = { my(f = factor(n)); prod(k=1, #f~, f[k, 2] < (2-(f[k, 1]<=3))); };

%Y Characteristic function of A276378.

%Y Absolute values and also parity of A355688.

%Y Cf. A008588, A008966, A354354.

%K nonn,easy,mult

%O 1

%A _Antti Karttunen_, May 28 2024