

A234334


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


5



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
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OFFSET

1,3


COMMENTS

Except a(1)=0, a(n) are numbers k such that both kx and yk 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



EXAMPLE

The two squares nearest to 25 are 16 and 25, because both 2525=0 and 2516=9 are squares, 25 is in the sequence.
The two squares nearest to 45 are 36 and 49, because both 4536=9 and 4945=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(na^2) and issqr((a+1)^2n)
end proc:


MATHEMATICA

filter[n_] := If[n == 0, True, Module[{a}, a = If[IntegerQ @ Sqrt[n], Sqrt[n]1, Floor[Sqrt[n]]]; IntegerQ @ Sqrt[na^2] && IntegerQ@Sqrt[(a+1)^2n]]];


CROSSREFS



KEYWORD

nonn,easy


AUTHOR



STATUS

approved



