Problems & Puzzles: Puzzles

Puzzle 1020. Follow-up to Puzzle 899 Carlos Rivera proposes the following follow-up to Puzzle 899:   In the Puzzle 899 Emmanuel Vantieghem improved his own solution given to Puzzle 897, about expressing the unit as a sum of 73035 Egyptian fractions using only denominators composed of 3 primes, instead of 73082 Egyptian fractions as before.   Here are all these 73035 denominators used. Q. Find a better solution for this target, using less than 73035 3-primes denominators.

---

Emmanuel Vantieghem wrote on November 14, 2020:

It is almost unbelievable, but the method I used to find a solution for  Problem 76 applied to Puzzle 1020 gives a serious amelioration of my previous result.

I now have a two set one with 70239 and another with 70189 numbers with three different prime factors whose reciprocals sum up to  1.
 

You can find the 70189 numbers solution in the annex.

***

Emmanuel Vantieghem wrote on November 26, 2020:

With some refinements of my method I succeeded in improving my previous record.

 

You will find  54642 squarefree numbers (each with three prime factors) whose reciprocals sum to 1 in the annex.

Later, on December 11, 2020, he sent another solution (file available on request) :

"with 54280 squarefree numbers (with three prime factors) whose reciprocals sum to  1... I think it is not minimal."

***