A066635 Distance from n to closest square different from n.

1, 1, 1, 3, 1, 2, 2, 1, 5, 1, 2, 3, 3, 2, 1, 7, 1, 2, 3, 4, 4, 3, 2, 1, 9, 1, 2, 3, 4, 5, 5, 4, 3, 2, 1, 11, 1, 2, 3, 4, 5, 6, 6, 5, 4, 3, 2, 1, 13, 1, 2, 3, 4, 5, 6, 7, 7, 6, 5, 4, 3, 2, 1, 15, 1, 2, 3, 4, 5, 6, 7, 8, 8, 7, 6, 5, 4, 3, 2, 1, 17, 1, 2, 3, 4, 5, 6, 7, 8, 9, 9, 8, 7, 6, 5, 4, 3, 2, 1, 19, 1, 2

Offset: 1

Author

Amarnath Murthy, Dec 29 2001

Keywords

• easy
• nonn
• look

Comments

The oscillatory nature of the sequence with increasing amplitude is quite evident.

Examples

			a(11) = 2 as 11 is closer to 9 than to 16 and 11 - 9 = 2.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Programs

Maple

N:= 10: # to get a(1)..a((N+1)^2-1)
seq(op([m^2-(m-1)^2, seq(i-m^2,i=m^2+1..m^2+m),seq((m+1)^2-i,i=m^2+m+1..(m+1)^2-1)]),m=1..N); # Robert Israel, Nov 07 2017

Mathematica

dncls[n_]:=If[IntegerQ[Sqrt[n]],Min[n-(Sqrt[n]-1)^2,(Sqrt[n]+1)^2-n],Min[n-Floor[Sqrt[n]]^2,Ceiling[Sqrt[n]]^2-n]]; Array[dncls,120] (* Harvey P. Dale, Dec 20 2024 *)

Formula

a(n) = Min {n - floor(sqrt(n))^2, (floor(sqrt(n))+1)^2 - n} if n is not a square. a(n) = 2*sqrt(n) - 1 if n is a square.

G.f.: -x^2/(1-x)^2 + Sum_{m>=1} x^(m^2)*(2m - x^m*(x+x^2)/(1-x)^2 -(1-4*x+x^2)/(1-x)^2). - Robert Israel, Nov 07 2017

Extensions

More terms from Larry Reeves (larryr(AT)acm.org), Apr 03 2002