[seqfan] Unit fractions, distinct or not

hv at crypt.org hv at crypt.org
Wed Nov 1 15:37:50 CET 2023

A few years ago I found a nice proof [1] that the least number of unit
fractions needed to sum to a rational q < 2 was the same whether or
not you required the unit fractions to be distinct. At the time
I considered also q > 2, but I could not prove that the distinct
requirement always needed more unit fractions in these cases, nor
could I find a counterexample.

This was prompted by a couple of sequences in which it wasn't clear
whether unit fractions were required to be distinct, and trying to
work out whether or when it mattered.

Looking at it again today, I quickly found this apparent counterexample:
  1 + 1/2 + 1/3 + 1/7 + 1/43 + 1/47
    = 171523/84882
    = 2 + 1759/84882
.. where it appears that 1759/84882 cannot be expressed as a sum of fewer
than 4 unit fractions (eg 1/49 + 1/3178 + 1/16859688 + 1/378998755750104).
So either last time or this time I made a mistake, and I'm not sure which.

Given sets S and multisets M of positive integers,
let d(q) := min(| S |): q = sum_{s \in S}{1/s}
and c(q) := min(| M |): q = sum_{m \in M}{1/m}.

I have two questions:
- am I correct that q = 2 + 1759/84882 has d(q) = c(q) = 6?
- can we say anything about the set of rationals q > 2 for which d(q) = c(q)?


[1] Repeatedly apply a splitting function 2/(2k+1) = 1/(k+1) + 1/(k+1)(2k+1)
to remove duplicates, and show that each step reduces the sum of a weight
function w(n) defined as the greatest power of 2 dividing n-1, and must
therefore terminate. This can show d(q) = c(q) for all rational 0 < q < 2.

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