Bounds for the Number of Terms of Harmonic Sums

This paper provides bounds for the number of terms, denoted by $f$, of a harmonic sum with the condition that it starts from any arbitrary unit fraction $\frac{1}{m}$, $m > 1$, until another unit fraction $\frac{1}{m+f-1}$ such that the sum is the highest sum less than a particular positive integer $q$. Also, we consider the number of terms of Egyptian fractions whose terms are consecutive multiples of $r$, $r \geq 1$, under the same above condition. We end the paper with a formula for the case: $q=1$ and $r=1$.