A053188 Distance from n to nearest square.

0, 0, 1, 1, 0, 1, 2, 2, 1, 0, 1, 2, 3, 3, 2, 1, 0, 1, 2, 3, 4, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 6, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 7, 7, 6, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 8, 7, 6, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4

Offset: 0

Author

Henry Bottomley, Mar 01 2000

Keywords

• easy
• nonn

Examples

			a(7)=2 since 9 is the closest square to 7 and |9-7| = 2.
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
• Index entries for sequences related to distance to nearest element of some set

Crossrefs

Cf. A053187, A074989, A000290.

Programs

Haskell

a053188 0 = 0
a053188 n = min (n - last xs) (head ys - n) where
   (xs,ys) = span (< n) a000290_list
-- Reinhard Zumkeller, Nov 28 2011

Mathematica

Flatten[Table[Abs[Nearest[Range[0,25]^2,n]-n],{n,0,120}]]  (* Harvey P. Dale, Mar 14 2011 *)

PARI

a(n)=abs(((sqrtint(4*n) + 1)\2)^2 - n) \\ Charles R Greathouse IV, Nov 16 2022

Python

from math import isqrt
def A053188(n): return abs(((m:=isqrt(n))+int(n-m*(m+1)>=1))**2-n) # Chai Wah Wu, Aug 03 2022

Formula

a(n) = |floor(sqrt(n) + 1/2)^2 - n|. - Ridouane Oudra, May 01 2019

a(n) <= sqrt(n). - Charles R Greathouse IV, Nov 16 2022