A158387 a(n) = -1 if n is a square, 1 if n is not a square.

-1, 1, 1, -1, 1, 1, 1, 1, -1, 1, 1, 1, 1, 1, 1, -1, 1, 1, 1, 1, 1, 1, 1, 1, -1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, -1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, -1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, -1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, -1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, -1

Offset: 1

Author

Jaroslav Krizek, Mar 17 2009

Keywords

• sign
• easy

Comments

Equivalently, a(n) is the sign of (-1)^[parity of number of divisors of n].

Examples

			a(12) = (-1)^6 = 1.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537
• Index entries for sequences computed from exponents in factorization of n.
• Index to divisibility sequences.

Crossrefs

Cf. A000005, A010052.

Cf. primes (A000040), pq = product of two distinct primes (A006881), pq...z = product of k (k > 2) distinct primes p, q, ..., z (A120944), p^k = prime powers (A000961(n) for n > 1), k = natural numbers (A000027).

Programs

Mathematica

Array[1 - 2 Boole[OddQ@ DivisorSigma[0, #]] &, 100] (* Michael De Vlieger, Nov 03 2017 *)
Table[If[IntegerQ[Sqrt[n]],-1,1],{n,120}] (* Harvey P. Dale, Feb 17 2020 *)

PARI

a(n) = (-1)^numdiv(n) \\ Michel Marcus, Jun 13 2013

PARI

a(n)=(-1)^issquare(n) \\ Charles R Greathouse IV, Jun 13 2013

PARI

first(n) = my(res = vector(n, i, -1)); for(i = 1, sqrtint(n), res[i^2] = 1); res \\ David A. Corneth, Nov 03 2017

Formula

a(n) = (-1)^tau(n) = (-1)^A000005(n).

a(1) = -1, a(p) = 1, a(pq) = 1, a(pq...z) = 1, a(p^k) = (-1)^(k+1), for p, q, ..., z primes.

Sum_{k=1..n} a(k) ~ n - 2*sqrt(n). - Amiram Eldar, Jan 13 2024

Extensions

Description corrected by David A. Corneth, Nov 03 2017