Consider the number $3600$. It is very special, because

$$\begin{array}{rlrl}3600& ={48}^2+& & {36}^2\\ 3600& ={20}^2+2\times & & {40}^2\\ 3600& ={30}^2+3\times & & {30}^2\\ 3600& ={45}^2+7\times & & {15}^2\end{array}$$

Similarly, we find that $88201={99}^2+{280}^2={287}^2+2\times {54}^2={283}^2+3\times {52}^2={197}^2+7\times {84}^2$.

In 1747, Euler proved which numbers are representable as a sum of two squares. We are interested in the numbers $n$ which admit representations of all of the following four types:

$$\begin{array}{rlrl}n& =a_1^2+& & b_1^2\\ n& =a_2^2+2& & b_2^2\\ n& =a_3^2+3& & b_3^2\\ n& =a_7^2+7& & b_7^2,\end{array}$$

where the $a_k$ and $b_k$ are positive integers.

There are $75373$ such numbers that do not exceed ${10}^7$.
How many such numbers are there that do not exceed $2\times {10}^9$?