### Finding certain partitions using Sage

### Context

Consider an integer $n \ge 1$.

A partition ~~$[p_1,p_2,...,p_n]$ with $2 \leq p_1 \leq ~~of $n$ is a list $p = [p_1, p_2, ..., p_k]$
of positive integers (called the parts) whose sum is $n$.

The list is considered sorted in nondecreasing order:
$p_1 \le p_2 ~~\leq ~~\le ... ~~\leq p_n$ and $n \geq 1$ ~~\le p_k$.

Consider an integer $d \ge 1$. The partition $p$
is called $d$-admissible ~~(where $d \geq 1$) ~~if ~~$n-d-1=\sum\limits_{i=1}^{n}{\frac{1}{p_i}}$.~~

all parts are at least $2$
and their inverses sum to $n - d - 1$. In formulas: $2 \le p_1$ and
$n - d - 1 = \sum \limits_{i=1}^{k} {\frac{1}{p_i}}$.

### Question

Is there a quick way to filter all partitions using ~~Sage ~~Sage
to find all $d$-admissible partitions for a fixed $d ~~\geq ~~\ge 1$?

Note that the assumptions imply $n ~~\leq 2(d+1)$ ~~\le 2(d+1)$
but the individual terms $p_i$ might get quite ~~large ~~large
(for $d=2$ the largest is already 42), which makes ~~it ~~it
very complicated to obtain a program that is ~~quick ~~quick
and works for large $d$.

For example for $d=1$ there are four 1-admissible ~~partitions, ~~partitions,
namely: [2,2,2,2], [3,3,3] ,[2,4,4] and ~~[2,3,6]. ~~[2,3,6].
For $d=2$ there are eighteen 2-admissible partitions.