(Almost)Consecutive-Integer Egyptian-Fraction Sum =1

[Leroy Quet]

[posted also to sci.math.]

I was led to this question by considering (m+1)-sided polygons where all 
but one of the sides are that of a k-gon, a (k+1)-gon, a (k+2)-gon,..., 
an (k+m)-gon.

But I wanted the final angle to be that of an n-gon, for some positive 
integer n.

For example,

1/3 + 1/4 + 1/5 + 1/6 + 1/20 = 1.

So, my question is,

which integers n are such that:

1/k + 1/(k+1) + 1/(k+2) +....+1/(k+m) + 1/n = 1 ?

What is the sequence of such n's? Is it in the EIS?

thanks,
Leroy Quet