A277425 a(n) = sqrt(16*t^2 - 32*t + k^2 + 8*k - 8*k*t + 16), where t = ceiling(sqrt(n)) and k = t^2 - n.

0, 2, 3, 4, 4, 5, 6, 7, 8, 6, 7, 8, 9, 10, 11, 12, 8, 9, 10, 11, 12, 13, 14, 15, 16, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28

Offset: 1

Author

Joseph Foley, Oct 14 2016

Keywords

• nonn
• easy

Comments

The equation 16*t^2 - 32*t + k^2 + 8*k - 8*k*t + 16 always produces a square, for any number n, with any t and k (i.e., t can be incremented and a corresponding k value is produced).

Examples

			n = 3, f(n) = 3; n = 11, f(n) = 7; n = 64, f(n) = 28; n = 103, f(n) = 22; n=208, f(n)= 39.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000

Programs

Magma

[n-Ceiling(Sqrt(n)-2)^2: n in [1..80]]; // Vincenzo Librandi, Nov 06 2016

Maple

seq(n-ceil(sqrt(n)-2)^2, n = 1 .. 64); # Ridouane Oudra, Jun 11 2019

Mathematica

Table[Function[t, Function[k, Sqrt[16 t^2 - 32 t + k^2 + 8 k - 8 k t + 16]][t^2 - n]]@ Ceiling@ Sqrt@ n, {n, 64}] (* or *)
Table[n - Ceiling[Sqrt[n] - 2]^2, {n, 64}] (* Michael De Vlieger, Nov 06 2016 *)

PARI

a(n) = n - (sqrtint(n-1)-1)^2 \\ Charles R Greathouse IV, Oct 14 2016

Formula

a(n) ~ 2*sqrt(n). - Charles R Greathouse IV, Oct 14 2016

a(n) = n - (floor(sqrt(n-1))-1)^2. - Charles R Greathouse IV, Oct 14 2016

a(n) = n - ceiling(sqrt(n) - 2)^2. - Vincenzo Librandi, Nov 06 2016