A243427 Floored (rational) values of sqrt(xy) such that sqrt(x) + sqrt(y) = sqrt(xy).

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, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66

Offset

1,1

Comments

(n + 1)^2/n = (n + 2 + 1/n) so for n>1 yields n+2 under the floor operation.

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.

Has solutions for all x<>1.

Links

Table of n, a(n) for n=1..64.

Formula

a(n) = floor((n + 1)^2/n).

a(n) = n + 2 for n>1.

Example

sqrt(4) + sqrt(4) = 4 = sqrt(16).
sqrt(9/4) + sqrt(9) = 4.5 = sqrt(81/4).

Prog

(JavaScript)
for (i = 1; i < 150; i++) {
document.write(Math.floor((i + 1) * (i + 1)/i) + ", ");
}

Crossrefs

Sequence in context: A202151 A368552 A120197 * A085581 A137902 A395263

Adjacent sequences: A243424 A243425 A243426 * A243428 A243429 A243430

Keyword

nonn,easy

Author

Jon Perry, Jun 05 2014

Status

approved