It depends on how you are going to define a sequence to be a $d$-sequence. For instance, in the Huneke's paper which I cited above, to define $d$-sequences, colon ideals are considered for any permutation of the sequence.

$(x^3):xy^2=(x^2)$, but $(x^3):x^2y^4=(x)$, therefore $x^3,xy^2$ is not a $d$-sequence.

@lemiller, Thanks. Would you please mention its reference? The question can be modifies as, whether $F$-pure ideals have a particular $F$-pure threshold, something like a function of the dimension of the ring or the codimension of the defining ideal

Let $k$ be a perfect field of characteristic $2$ and $R=\bigcup^{\infty} k[[x^{1/2^i}]]$. Then the unique maximal ideal $\mathfrak{m}$ of $R$ is generated by $\{x^{1/2^i}:i\in \mathbb{N}\}$. It is easily seen that $\mathfrak{m}^2=\mathfrak{m}$. In general if $R$ is a perfect ring of prime characteristic $p>0$ then any prime ideal $\mathfrak{p}$ of $R$ satisfies $\mathfrak{p}^2=\mathfrak{p}$, whence $\mathfrak{p}\mathfrak{m}=\mathfrak{p}$, provided $(R,\mathfrak{m})$ is local as well.

@Lau-tzu $Hom_{\mathbb{Z}}(\mathbb{Q},\mathbb{Z})=0$

@მამუკა ჯიბლაძე, how do we define addition on an affine space? do you mean $Spec(K[x_1,\ldots,x_n])$ by an affine space over a field $K$?