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

1,3

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.

Links

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

Example

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.

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 *)

Crossrefs

Cf. A000290.

Cf. subsequences: A007204, A234335, A234336.

Cf. A234348.

Sequence in context: A375350 A166652 A286571 * A025623 A394592 A069959

Adjacent sequences: A234331 A234332 A234333 * A234335 A234336 A234337

Keyword

nonn,easy

Author

Alex Ratushnyak, Dec 23 2013

Status

approved