Lorentz-friend numbers

n and m natural numbers Lorentz-friends if the Lorentz-transformation of the two Egyptian fractions 1/n and 1/m is also an Egyptian fraction.

As a reminder: Lorentz-transformation of n and m is (n+m)/(+nm).

So n and m are Lorentz-friend if and only if exists natural number k that

(1/n+1/m)/(1+1/nm) = 1/k

Like the light, 1 is Lorentz-friend with all naturals (the Lorentz transformation of 1 and m is 1)! There is concern about this kind of indiscriminate friendship, so we would rather call it pseudo-friendship and ignore it simply when talking about Lorentz friendship.

A perfect antipode is 2 that has no true Lorentz friends.

Another extremely interesting phenomenon is that every odd number and the following odd are Lorentz friends. In fact:

(1/2n-1)+1/(2n+1))/(1+1/(2n-1)(2n+1)) = 1/n

It follows from the above that every odd number greater than 3 has at least two Lorentz friends.

Is it possible to know how many Lorentz friends a natural number has? What relationships and characteristics can be established?

Every natural number greater than 2 seems to have at least one, every odd number to has at least two Lorentz friends, but not always more. However, there are many examples where a number has 3 or more Lorentz friends.

For example 16 have 3 friends: 35, 69 and 239. The nearby 18 only have one friend: 305.

There are bigger achievements. So 31 have 13 friends: 9, 17, 29, 33, 49, 65, 89, 129, 161, 209, 289, 449, 929.

We can define a sequence a(n) that gives n the smallest Lorentz friend which largest that n. Clearly, if n is odd a(n)=n+2, it can make the sequence a bit dull. Let's hope that some interesting things will come. For the sake of order, it should be agreed that a(1)=1, and if n has no friends, then a(n)=0. So the sequence looks like this:

1, 0, 5, 11, 7, 29, 9, 13, 11, 23, 13, 131, 15, 25, 17, 35, 19, 305, 21, 37, 23,…

To be continued.

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