A302706 - OEIS

%I #25 Oct 23 2018 03:04:41

%S 0,2,3,4,5,6,10,11,12,13,14,16,18,18,18,26,27,28,29,30,32,32,33,34,35,

%T 40,40,40,50,51,52,53,54,55,56,57,58,59,60,61,62,63,72,72,72,82,83,84,

%U 85,86,87,88,89,90,91,92,98,98,98,98,98,98,99,100,104,104,122,123,124,125,126,127,128,129,130,131,132

%N a(n) is the maximum remainder of x^2 + y^2 divided by x + y with 0 < x <= y <= n.

%C Values of a(n) such that a(n) is prime are 2, 3, 5, 11, 13, 29, 53, 59, 61, 83, 89, 127, 131, 137, 139, 173, ...

%C Conjecture: lim_{n->inf} a(n)/(2n) = 1, with both variables x and y taking values asymptotically close to n. - _Andres Cicuttin_, Oct 18 2018

%H Altug Alkan, <a href="/A302706/b302706.txt">Table of n, a(n) for n = 1..1000</a>

%e a(1) = 0 because x = y = 1 is only option.

%e a(13) = a(14) = a(15) = 18 because (7^2 + 13^2) mod (7 + 13) = 18 is the largest corresponding remainder for them.

%t a[n_]:=Table[Table[Mod[x^2+y^2 ,x+y],{x,1,y}],{y,1,n}]//Flatten//Max;

%t Table[a[n],{n,1,100}]

%o (PARI) a(n) = vecmax(vector(n, x, vecmax(vector(x, y, (x^2+y^2) % (x+y))))); \\ after _Michel Marcus_ at A302245

%Y Cf. A053626, A302245.

%K nonn,easy

%O 1,2

%A _Altug Alkan_ and _Andres Cicuttin_, Apr 12 2018