I've recently came across an interesting paper : ON FINITE SUMS OF RECIPROCALS OF DISTINCT n-th POWERS by R. L. GRAHAM. At the end of the paper he gives some exemples:
$$ \frac{1}2 = \frac1{2^2}+\frac1{3^2}+\frac1{4^2}+\frac1{5^2}+\frac1{6^2}+\frac1{15^2}+\frac1{18^2}+\frac1{36^2}+\frac1{60^2}+\frac1{180^2}$$
I've tried my best to make a program to find that result but i can't find a solution.
I found this [solution](https://math.stackexchange.com/questions/1921652/can-every-number-be-represented-as-a-sum-of-different-reciprocal-numbers) to an easier problem : the representation of a fraction as the sum of distinct reciprocals. I tried to expend this algorithm to solve my problem but when i execute my program the numbers given get too large:

```python
def pgcd(a,b):
    return a if b==0 else pgcd(b,a%b)
def subFrac(p1,q1,p2,q2):
    u,v=p1*q2-p2*q1,q1*q2
    d=pgcd(u,v)
    return (u//d,v//d)
def dec(p,q):
    print(p,q)
    frac=[]
    k=2
    while p/q>1/(k**2):
        p,q=subFrac(p,q,1,k**2)
        frac.append(k)
        print(k,p,q)
        k+=1
    while p>0:
        while 1/(k**2)>p/q:
            k+=1
        p,q=subFrac(p,q,1,k**2)
        print(k,p,q)
        frac.append(k)
        k+=1
    return frac
```
Gives :
$$ \frac{1}2 = \frac1{2^2}+\frac1{3^2}+\frac1{4^2}+\frac1{5^2}+\frac1{6^2}+\frac1{11^2}+\frac1{54^2}+\frac1{519^2}+\frac1{59429^2}+...$$
In general, I'd like to make a program that gives a representation with reasonable number like the one given or this one from Wikipedia:
$$ \frac{1}2 = \frac1{2^2}+\frac1{3^2}+\frac1{4^2}+\frac1{5^2}+\frac1{7^2}+\frac1{12^2}+\frac1{15^2}+\frac1{20^2}+\frac1{28^2}+\frac1{35^2}$$
Thanks in advance!