sum of unit fractions

[hv at crypt.org]

hv at crypt.org wrote:
:hv at crypt.org wrote:
::Define a(n) as the least integer k such that there is a sum of distinct unit
::fractions equal to _n_ of which the greatest denominator is k.
::
::Alternatively:
::  a(n) = k implies that there exists a set of positive integers S such that
::  sum{1/s_i} = n, and max(S) = k, and no set S' exists with the same sum
::  but a smaller maximal element.
::
::eg a(1) = 1 (S = { 1 })
::   a(2) = 6 (S = { 1 2 3 6 })
::   a(3) = 24 (S = { 1 2 3 4 5 6 8 9 10 15 18 20 24 })
:[...]
::Working by hand, I believe I've established a(4) = 65 [...]
:My best attempt for a(5) is 184
:.. and for a(6) is 475
:My figures for a(4) .. a(6) should be viewed as best known upper bounds
:unless and until someone can confirm them.

A new program that exhaustively tries all possibilities confirms these
values as optimal up to a(5); it established a(6) > 468 before running
out of memory, but it seems that my original greedy algorithm did well
at finding optimal solutions even without backtracking.

I'll keep working on it ...

Hugo