Context: For this X-max, I will make $50$ presents by myself for my family. I would like to label them so that the labels are totally ordered and all different. However, I think that the traditional $1,2,3,\ldots,50$ is boring.
So, I tried to find a more "funny" way to numerate them and came with an approach based on the representation of $\pi$. After a bit of googling, it turns out that the sequence of labels I'd like to use is the sequence A064809: "Decimal expansion of $\pi$ written as a sequence of natural numbers avoiding duplicates."
I will denote this sequence by $(x_k)_{k\in\Bbb N}$.
The $12$ first terms of this sequence: $$ \large \pi = \color{red}3.\color{blue}1\color{green}4\color{pink}{1 5}\color{yellow}9\color{grey}2\color{maroon}6\color{teal}5\color{olive}{3 5}\color{purple}8\color{fuchsia}{9 7}\color{aqua}{93}$$ so that $(x_1,x_2,\ldots,x_{12})=(3,1,4,15,9,2,6,5,35,8,97,93)$.
My question is the following:
Let $S=\{x_1,x_2,\ldots\}$ and $x\in S$. Is there an efficient way to find $i$ such that $x_i=x$?
By efficient, I mean that we don't need to compute all the first terms of $(x_k)$ until one detects $x$.