A379971 a(n) = 1 if n^2-1 is squarefree, otherwise 0.

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

Offset

1

Links

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

Index entries for characteristic functions.

Formula

a(1) = 0; for n > 1, a(n) = A008966(n^2-1).

a(n) = A323239(n-1) * A323239(n+1).

Sum_{k=1..n} a(k) ~ c*n, where c = Product_{p prime} (1 - 2/p^2) = 0.322634... (A065474). - Amiram Eldar, Jan 22 2025

Mathematica

a[n_] := Boole[SquareFreeQ[n^2-1]]; Array[a, 100] (* Amiram Eldar, Jan 22 2025 *)

Prog

(PARI) A379971(n) = issquarefree(n^2-1);
(Python)
from sympy import factorint
def A379971(n): return 0 if n<=1 else int(max((a:=factorint(n-1)).values(), default=0)<=1 and max((b:=factorint(n+1)).values(), default=0)<=1 and not set(a)&set(b)) # Chai Wah Wu, Jan 22 2025

Crossrefs

Characteristic function of A067874.

Cf. A005117, A008966, A065474, A323239.

Sequence in context: A141743 A268384 A358670 * A288524 A112416 A061265

Adjacent sequences: A379968 A379969 A379970 * A379972 A379973 A379974

Keyword

nonn

Author

Antti Karttunen, Jan 22 2025

Status

approved