A049240 Smallest nonnegative value taken on by x^2 - n*y^2 for an infinite number of integer pairs (x, y).

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

Offset: 1

Author

David W. Wilson

Keywords

• nonn
• easy

Comments

Encodes to 1,2,1,4,1,6,1,8,1,10,...: unsigned version of A009531. - Paul Barry, Oct 12 2005

Parity of inverse Moebius transform of Jacobsthal numbers J(k) less J(n). - Paul Barry, Oct 12 2005

Links

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

Crossrefs

Cf. A001045, A005931, A010052.

Characteristic function of A000037 (the nonsquares).

Programs

Magma

[Floor((1 + Ceiling(Sqrt(n)) - Floor(Sqrt(n)))/2) : n in [1..100]]; // Wesley Ivan Hurt, Sep 27 2014

Maple

A049240:=n->`if`(issqr(n),0,1): seq(A049240(n), n=1..100); # Wesley Ivan Hurt, Sep 27 2014

Mathematica

Differences[Table[n - Ceiling[Sqrt[n]], {n, 105}]] (* Arkadiusz Wesolowski, Oct 30 2012 *)
Table[Floor[(1 + Ceiling[Sqrt[n]] - Floor[Sqrt[n]])/2], {n, 70}] (* Wesley Ivan Hurt, Sep 27 2014 *)

Python

from math import isqrt
def A049240(n): return int(isqrt(n)**2!=n) # Chai Wah Wu, Jun 14 2022

Formula

a(n) = 0 if n is square, 1 otherwise.

a(n) = (A001045(n) - Sum_{k|n} A001045(k)) mod 2. - Paul Barry, Oct 12 2005

a(n) = 1 - A010052(n). - R. J. Mathar, Jul 04 2009

a(n) = floor(1+ceiling(sqrt(n))-floor(sqrt((n)))/2). - Wesley Ivan Hurt, Sep 27 2014

G.f.: (1+x)/(2-2*x) - (1/2)*theta_3(0,x) where theta_3 is a Jacobi theta function. - Robert Israel, Oct 02 2014