Nested square roots

Published on Saturday, 7th January 2017, 07:00 pm; Solved by 105;
Difficulty rating: 100%

Problem 585

Consider the term $\small \sqrt{x+\sqrt{y}+\sqrt{z}}$ that is representing a nested square root. $x$, $y$ and $z$ are positive integers and $y$ and $z$ are not allowed to be perfect squares, so the number below the outer square root is irrational. Still it can be shown that for some combinations of $x$, $y$ and $z$ the given term can be simplified into a sum and/or difference of simple square roots of integers, actually denesting the square roots in the initial expression.

Here are some examples of this denesting:
$\small \sqrt{3+\sqrt{2}+\sqrt{2}}=\sqrt{2}+\sqrt{1}=\sqrt{2}+1$
$\small \sqrt{8+\sqrt{15}+\sqrt{15}}=\sqrt{5}+\sqrt{3}$
$\small \sqrt{20+\sqrt{96}+\sqrt{12}}=\sqrt{9}+\sqrt{6}+\sqrt{3}-\sqrt{2}=3+\sqrt{6}+\sqrt{3}-\sqrt{2}$
$\small \sqrt{28+\sqrt{160}+\sqrt{108}}=\sqrt{15}+\sqrt{6}+\sqrt{5}-\sqrt{2}$

As you can see the integers used in the denested expression may also be perfect squares resulting in further simplification.

Let F($n$) be the number of different terms $\small \sqrt{x+\sqrt{y}+\sqrt{z}}$, that can be denested into the sum and/or difference of a finite number of square roots, given the additional condition that $0<x \le n$. That is,
$\small \displaystyle \sqrt{x+\sqrt{y}+\sqrt{z}}=\sum_{i=1}^k s_i\sqrt{a_i}$
with $k$, $x$, $y$, $z$ and all $a_i$ being positive integers, all $s_i =\pm 1$ and $x\le n$.
Furthermore $y$ and $z$ are not allowed to be perfect squares.

Nested roots with the same value are not considered different, for example $\small \sqrt{7+\sqrt{3}+\sqrt{27}}$, $\small \sqrt{7+\sqrt{12}+\sqrt{12}}$ and $\small \sqrt{7+\sqrt{27}+\sqrt{3}}$, that can all three be denested into $\small 2+\sqrt{3}$, would only be counted once.

You are given that F(10)=17, F(15)=46, F(20)=86, F(30)=213 and F(100)=2918 and F(5000)=11134074.
Find F(5000000).