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Let an 'Egyptian unity sum set' be a set of positive integers {a, b, c ...} such that their Egyptian fractions sum to 1; and none of the elements are equal. That is:

1/a + 1/b + 1/c ... = 1

Let the number of elements in any such set be equal to N. For any such set, let the a+b+c... = Z

For instance, with set {2,3,6} N=3, Z=11

Are there any two 'Egyptian unity sum sets' which have the same Z value?

I haven't found any yet, but I suspect they would exist. Are there any pairs which have the same N value too?

Which integers cannot be a Z value for any set?

Z=11 is the minimum value, and Z= 24 seems to be the next smallest (for set {2,4,6,12}); so there are plenty of these 'not-Z' integers initially. Are there an infinite number of them in total; or is there some value above which all integers are Z-values for some set?

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    $\begingroup$ $2+3+11+22+33=2+5+8+12+20+24$ $\endgroup$ Aug 3, 2015 at 6:56
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    $\begingroup$ For also same N value: $2+4+9+12+18=2+5+6+12+20$ $\endgroup$ Aug 3, 2015 at 7:03
  • $\begingroup$ Another example 4+7+8+9+12+14+18+24+27+28+36+54=4+5+7+8+10+12+15+40+140 $\endgroup$
    – Yuriy S
    Feb 2, 2017 at 13:56

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For your first question the answer is yes, such examples i gave in the comments. As for your second question, in this OEIS it is stated that "R. L. Graham showed that a(n)>0 for n>77", where a(n) is the number of ways to express 1 as the sum of distinct unit fractions such that the sum of the denominators is n.

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