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 $t(n)\in \mathrm{\Omega }({\log }_2(n))\cap O(\sqrt{n})$. Explicit lower and upper bounds for $t(n)$ will be provided for some small integer $n$.