A053866 - OEIS

%I #76 Nov 03 2024 09:32:32

%S 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,

%T 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,

%U 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

%N 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.

%C Also parity of A001227, the number of odd divisors of n. - _Omar E. Pol_, Apr 04 2016

%C Also parity of A000593, the sum of odd divisors of n. - _Omar E. Pol_, Apr 05 2016

%C Characteristic function of A028982. - _Antti Karttunen_, Sep 25 2017

%C It appears that this is also the parity of A067742, the number of middle divisors of n. - _Omar E. Pol_, Mar 18 2018

%C Also parity of the deficiency of n (A033879) and of the abundance of n (A033880). - _Omar E. Pol_, Nov 02 2024

%H Antti Karttunen, <a href="/A053866/b053866.txt">Table of n, a(n) for n = 1..65536</a>

%H J. N. Cooper and A. W. N. Riasanovsky, <a href="http://www.math.sc.edu/~cooper/Sigma.pdf">On the Reciprocal of the Binary Generating Function for the Sum of Divisors</a>, 2012.

%H J. N. Cooper and A. W. N. Riasanovsky, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Cooper/cooper3.html">On the Reciprocal of the Binary Generating Function for the Sum of Divisors</a>, J. Int. Seq. 16 (2013) #13.1.8.

%H Michael Gilleland, <a href="/selfsimilar.html">Some Self-Similar Integer Sequences</a>

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

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

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

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

%F Dirichlet g.f.: zeta(2s)(1+2^-s). - _Michael Somos_, Apr 12 2004

%F a(n) = A001157(n) mod 2. - _R. J. Mathar_, Apr 02 2011

%F a(n) = floor(sqrt(n)) + floor(sqrt(n/2)) - floor(sqrt(n-1))-floor(sqrt((n-1)/2)). - _Enrique Pérez Herrero_, Oct 15 2013

%F a(n) = A000035(A000203(n)). - _Omar E. Pol_, Oct 26 2013

%F a(n) = A063524(A286357(n)) = A063524(A292583(n)). - _Antti Karttunen_, Sep 25 2017

%F a(n) = A295896(A156552(n)). - _Antti Karttunen_, Dec 02 2017

%F a(n) = Sum_{ m: m^2|n } A019590(n/m^2). - _Andrey Zabolotskiy_, May 07 2018

%F G.f.: (theta_3(x) + theta_3(x^2))/2 - 1. - _Ilya Gutkovskiy_, May 23 2019

%F Sum_{k=1..n} a(k) ~ (1 + 1/sqrt(2)) * sqrt(n). - _Vaclav Kotesovec_, Oct 16 2020

%p A053866:= (n -> numtheory[sigma](n) mod 2):

%p seq (A053866(n), n=0..104); # _Jani Melik_, Jan 28 2011

%t Mod[DivisorSigma[1,Range[110]],2] (* _Harvey P. Dale_, Sep 04 2017 *)

%o (PARI) {a(n) = if( n<1, 0, issquare(n) || issquare(2*n))} /* _Michael Somos_, Apr 12 2004 */

%o (Python)

%o from sympy.ntheory.primetest import is_square

%o def A053866(n): return int(is_square(n) or is_square(n<<1)) # _Chai Wah Wu_, Jan 09 2023

%Y Essentially same as A093709.

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

%K nonn,mult

%O 1,1

%A _Henry Bottomley_, Mar 29 2000

%E More terms from _James A. Sellers_, Apr 08 2000

%E Alternative description added to the name by _Antti Karttunen_, Sep 25 2017