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%I #20 May 14 2017 13:19:12
%S 4,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,
%T 28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,
%U 51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66
%N Floored (rational) values of sqrt(xy) such that sqrt(x) + sqrt(y) = sqrt(xy).
%C (n + 1)^2/n = (n + 2 + 1/n) so for n>1 yields n+2 under the floor operation.
%C Consider sqrt(x) + sqrt(y) = sqrt(xy). If we let y=d^2*x, then the LHS becomes (1+d)sqrt(x) and the RHS becomes dx. Divide both sides by sqrt(x), take the d from the RHS to the LHS and square giving x=((1+d)/d)^2, and so y is (1+d)^2 and the original RHS is now (1+d)^2/d. This sequence is concerned with d being an integer.
%C Has solutions for all x<>1.
%F a(n) = floor((n + 1)^2/n).
%F a(n) = n + 2 for n>1.
%e sqrt(4) + sqrt(4) = 4 = sqrt(16).
%e sqrt(9/4) + sqrt(9) = 4.5 = sqrt(81/4).
%o (JavaScript)
%o for (i = 1; i < 150; i++) {
%o document.write(Math.floor((i + 1) * (i + 1)/i) + ", ");
%o }
%K nonn,easy
%O 1,1
%A _Jon Perry_, Jun 05 2014
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