It is well known that certain sums of reciprocals add up to $2$:
- the reciprocals of powers of $2$:
$$
{1 \over 1}+{1 \over 2}+{1 \over 4}+{1 \over 8}+{1 \over 16}+{1 \over 32}+\cdots =2,
$$
- the reciprocals of the triangular numbers:
$$
{1 \over 1}+{1 \over 3}+{1 \over 6}+{1 \over 10}+{1 \over 15}+{1 \over 21}+\cdots =2,
$$
- $\ldots$
Are there any sums such that $\sum_{i=0}^{\infty} s_i^{-1}=2$ for which all the $s_i$ are all distinct odd positive integers?
Similarly, are there any sums such that $\exists k\in\mathbb{Z}, k>0$ such that $\sum_{i=0}^{k} s_i^{-1}=2$ for which all the $s_i$ are all distinct odd positive integers?
Best Answer
If you allow $\frac{1}{1}$, which you seem to, judging by your question:
Try adding the reciprocals of $1, 3, 5, 7, 9, 15, 21, 27, 35, 63, 105, 135$.
Found by choosing an odd abundant number, namely $945$ and finding a sum of factors that adds to $2\cdot 945$:
$945+315+189+135+105+63+45+35+27+15+9+7=2\cdot 945$.
Then divide this equation by $945$, reducing the individual fractions on the left.