It is well known that every positive integer is the sum of at most four perfect squares (including $1$).
But which positive integers are not the sum of four non-zero perfect squares ($1$ is still allowed as a perfect square) ?
I showed that the numbers $2^k$ , $2^k\cdot 3$ and $2^k\cdot 7$ with odd positive integer $k$ have this property. I checked the numbers upto $10^4$ and above $41$, no examples , other than those of the mentioned forms , occured. So my question is whether additional positive integers with the desired property exist.