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

Author

Henry Bottomley, Mar 29 2000

Keywords

• nonn
• mult

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

Also parity of the deficiency of n (A033879) and of the abundance of n (A033880). - Omar E. Pol, Nov 02 2024

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

Crossrefs

Essentially same as A093709.

Cf. A000203, A000593, A001227, A028982, A033879, A033880, A286357, A292583, A295896, A019590, A025441, A145393, A359548 (Dirichlet inverse).

Programs

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 *)

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

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

Extensions

More terms from James Sellers, Apr 08 2000

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