A068527 Difference between smallest square >= n and n.

0, 0, 2, 1, 0, 4, 3, 2, 1, 0, 6, 5, 4, 3, 2, 1, 0, 8, 7, 6, 5, 4, 3, 2, 1, 0, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9

Offset: 0

Author

Vladeta Jovovic, Mar 21 2002

Keywords

• nonn
• easy

Comments

The greedy inverse (sequence of the smallest k such that a(k)=n) starts 0, 3, 2, 6, 5, 11, 10, 18, 17, 27, 26, 38, 37, 51, 50, ... and appears to be given by A010000 and A002522, interleaved. - R. J. Mathar, Nov 17 2014

Links

• Ivan Panchenko, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A053186, A068869, A066857.

Bisections: A348596, A350962.

Programs

Magma

[ Ceiling(Sqrt(n))^2-n : n in [0..50] ]; // Wesley Ivan Hurt, Jun 11 2014

Maple

A068527:=n->ceil(sqrt(n))^2-n; seq(A068527(n), n=0..100); # Wesley Ivan Hurt, Jun 11 2014

Mathematica

Table[Ceiling[Sqrt[n]]^2-n,{n,0,100}] (* Vladimir Joseph Stephan Orlovsky, Feb 01 2012 *)

PARI

a(n)=if(issquare(n), 0, (sqrtint(n)+1)^2-n) \\ Charles R Greathouse IV, Oct 22 2014

Python

from math import isqrt
def A068527(n): return 0 if n == 0 else (isqrt(n-1)+1)**2-n # Chai Wah Wu, Feb 22 2022

Formula

a(n) = A048761(n) - n = ceiling(sqrt(n))^2 - n.

G.f.: (-x^2 + (x-x^2)*Sum_{m>=1} (1+2*m)*x^(m^2))/(1-x)^2. This sum is related to Jacobi Theta functions. - Robert Israel, Nov 17 2014