A234334 Numbers k such that both distances from k to two nearest squares are perfect squares.

0, 1, 5, 8, 25, 40, 45, 65, 80, 153, 160, 169, 200, 221, 325, 360, 416, 425, 493, 520, 625, 680, 725, 925, 936, 1025, 1040, 1073, 1088, 1305, 1360, 1681, 1768, 1800, 1813, 1845, 1961, 2000, 2320, 2385, 2501, 2600, 2925, 3016, 3185, 3200, 3400, 3445, 3721, 3848

Offset: 1

Author

Alex Ratushnyak, Dec 23 2013

Keywords

• nonn
• easy

Comments

Except a(1)=0, a(n) are numbers k such that both k-x and y-k are perfect squares, where x and y are two nearest to k squares: x < k <= y.

The sequence of sums of distances begins: 1, 1, 5, 5, 9, 13, 13, 17, 17, 25, 25, 25, 29, 29, 37, 37, 41, 41, 45, 45, 49, 53, 53, 61, 61, 65, 65, 65, 65, 73, 73, 81, 85, ... (cf. A057653).

Each term is either a square or has a pair: if i^2 + j^2 = 2*m+1 then m^2+i^2 and m^2+j^2 are both in the sequence.

Examples

			The two squares nearest to 25 are 16 and 25, because both 25-25=0 and 25-16=9 are squares, 25 is in the sequence.
The two squares nearest to 45 are 36 and 49, because both 45-36=9 and 49-45=4 are squares, 45 is in the sequence.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..6000

Crossrefs

Cf. A000290.

Cf. subsequences: A007204, A234335, A234336.

Cf. A234348.

Programs

Maple

filter:= proc(n) local a;
  if issqr(n) then a:= sqrt(n)-1 else a:= floor(sqrt(n)) fi;
  issqr(n-a^2) and issqr((a+1)^2-n)
end proc:
select(filter, [$0..5000]); # Robert Israel, Jan 21 2021

Mathematica

filter[n_] := If[n == 0, True, Module[{a}, a = If[IntegerQ @ Sqrt[n], Sqrt[n]-1, Floor[Sqrt[n]]]; IntegerQ @ Sqrt[n-a^2] && IntegerQ@Sqrt[(a+1)^2-n]]];
Select[Range[0, 5000], filter] (* Jean-François Alcover, May 28 2023, after Robert Israel *)