Tight universal quadratic forms

For a positive integer $n$, let $\mathcal T(n)$ be the set of all integers greater than or equal to $n$. An integral quadratic form $f$ is called tight $\mathcal T(n)$-universal if the set of nonzero integers that are represented by $f$ is exactly $\mathcal T(n)$. The smallest possible rank over all tight $\mathcal T(n)$-universal quadratic forms is defined by $t(n)$. In this article, we find all tight $\mathcal T(n)$-universal diagonal quadratic forms. We also prove that $t(n) \in \Omega(\log_2(n)) \cap O(\sqrt{n})$. Explicit lower and upper bounds for $t(n)$ will be provided for some small integer $n$.