A053866 Parity of A000203(n), the sum of the divisors of n; a(n) = 1 when n is a square or twice a square, 0 otherwise.

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

Offset

1,1

Comments

Also parity of A001227, the number of odd divisors of n. - Omar E. Pol, Apr 04 2016

Also parity of A000593, the sum of odd divisors of n. - Omar E. Pol, Apr 05 2016

Characteristic function of A028982. - Antti Karttunen, Sep 25 2017

It appears that this is also the parity of A067742, the number of middle divisors of n. - Omar E. Pol, Mar 18 2018

Links

Antti Karttunen, Table of n, a(n) for n = 1..65536

J. N. Cooper and A. W. N. Riasanovsky, On the Reciprocal of the Binary Generating Function for the Sum of Divisors, 2012.

J. N. Cooper and A. W. N. Riasanovsky, On the Reciprocal of the Binary Generating Function for the Sum of Divisors, J. Int. Seq. 16 (2013) #13.1.8.

Michael Gilleland, Some Self-Similar Integer Sequences

Index entries for characteristic functions

Formula

a(n) = A000203(n) mod 2. a(n)=1 iff n>0 is a square or twice a square.

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

a(n) = A093709(n) if n>0.

Dirichlet g.f.: zeta(2s)(1+2^-s). - Michael Somos, Apr 12 2004

a(n) = A001157(n) mod 2. - R. J. Mathar, Apr 02 2011

a(n) = floor(sqrt(n)) + floor(sqrt(n/2)) - floor(sqrt(n-1))-floor(sqrt((n-1)/2)). - Enrique Pérez Herrero, Oct 15 2013

a(n) = A000035(A000203(n)). - Omar E. Pol, Oct 26 2013

a(n) = A063524(A286357(n)) = A063524(A292583(n)). - Antti Karttunen, Sep 25 2017

a(n) = A295896(A156552(n)). - Antti Karttunen, Dec 02 2017

a(n) = Sum_{ m: m^2|n } A019590(n/m^2). - Andrey Zabolotskiy, May 07 2018

G.f.: (theta_3(x) + theta_3(x^2))/2 - 1. - Ilya Gutkovskiy, May 23 2019

Sum_{k=1..n} a(k) ~ (1 + 1/sqrt(2)) * sqrt(n). - Vaclav Kotesovec, Oct 16 2020

Maple

A053866:= (n -> numtheory[sigma](n) mod 2):
seq (A053866(n), n=0..104); # Jani Melik, Jan 28 2011

Mathematica

Mod[DivisorSigma[1, Range[110]], 2] (* Harvey P. Dale, Sep 04 2017 *)

Prog

(PARI) {a(n) = if( n<1, 0, issquare(n) || issquare(2*n))} /* Michael Somos, Apr 12 2004 */
(Python)
from sympy.ntheory.primetest import is_square
def A053866(n): return int(is_square(n) or is_square(n<<1)) # Chai Wah Wu, Jan 09 2023

Crossrefs

Essentially same as A093709.

Cf. A000203, A000593, A001227, A028982, A286357, A292583, A295896, A019590, A025441, A145393.

Sequence in context: A266459 A354199 A214509 * A143259 A359818 A207710

Adjacent sequences: A053863 A053864 A053865 * A053867 A053868 A053869

Keyword

nonn,mult

Author

Henry Bottomley, Mar 29 2000

Extensions

More terms from James A. Sellers, Apr 08 2000

Alternative description added to the name by Antti Karttunen, Sep 25 2017

Status

approved