How to shorten Python code involving the sum of reciprocals?

Question

I have a coding problem that goes like this:

Given a positive integer \$N\$, return a list of all possible pairs of positive integers \$(x,y)\$ such that $$\frac1x+\frac1y=\frac1N$$

I already solved the problem using Python, but I was wondering how I can code golf it. The following is my attempt at golfing (I try to follow some tips on the tips page):

108 bytes

exec("f=lambda N:[b for i in range(1,N+1)if N*N%i==0"+"for b in[%s,%s[::-1]]][:-1]"%(("[N+i,N+N*N//i]",)*2))

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Ungolfed code for better readability

def f(N):
 l=[]
 for i in range(1,N+1):
  if N*N%i==0:
   a=[N+i,N+N*N//i]
   l.extend([a,a[::-1]])
 return l[:-1]

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Answer 1

Python 3.8 (pre-release), 60 bytes

f=lambda N:[[N+i,N+N*N//i]for i in range(1,N*N+1)if 1>N*N%i]

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No need for reversing:

We just generate every fraction for which the first denominator is a divisor of N squared.

Answer 2

As AnttiP has pointed out, this can be shorter with a change of approach and full rewrite. Let's look at some simpler manipulations of your original code though just for the sake of it though:

Here was my first attempt at eliminating the exec hack:

85 bytes

lambda N:[b for i in range(1,N+1)if N*N%i==0for b in[A:=[N+i,N+N*N//i],A[::-1]]][:-1]

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Unfortunately, as you may notice, it doesn't work, because := is not allowed in for comprehension iterable expression.

There's another place we can put it in though: the if clause just before it:

91 bytes

lambda N:[b for i in range(1,N+1)if(A:=[N+i,N+N*N//i])and N*N%i==0for b in[A,A[::-1]]][:-1]

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This can be shortened a bit by combining the comparison ==0:

89 bytes

lambda N:[b for i in range(1,N+1)if 0in(A:=[N+i,N+N*N//i],N*N%i)for b in[A,A[::-1]]][:-1]

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The [:-1] at the end is just to remove the final duplicated result of (x, x), but we can remove this just by switching to a set comprehension (and using a tuple instead of a list, because it's hashable):

84 bytes

lambda N:{b for i in range(1,N+1)if 0in(A:=(N+i,N+N*N//i),N*N%i)for b in[A,A[::-1]]}

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The output is no longer in order, so hopefully that doesn't matter here.

I'm working on going further than this... stay tuned