I have the following problem :
For a given positive real $A>1$
$$A = \sum_{i=1}^{\infty} \frac{i}{a_i}$$
$$1 = \sum_{i=1}^{\infty} \frac{1}{a_i}$$
$$a_{n+1}>a_{n}$$
where $a_i$ is a strictly increasing sequence of strict positive integers.
How to find $a_i$ for a given closed form $A$, in particular in some kind of closed form ?
I was thinking about continued fractions , odd greedy expansions, engel expansions, fractal fractions and such things.
Is the solution even unique ??
See also :
I want to add that there exists a lower bound for $A$ IF $a_1 = 2$ that is larger than $1$ and a bound for the rate of $a_i$. This is explained by
$$B = \sum_{i=1}^{\infty} \frac{i}{b_i}$$
$$1 = \sum_{i=1}^{\infty} \frac{1}{b_i}$$
$$b_{n+1}>b_{n}$$
Where $b_i$ equals to Sylvester's sequence $2,3,7,43,1807,...$, the sum of its (Sylvester's sequence) forms a series of unit fractions that converges to 1 more rapidly than any other series of unit fractions.
See : https://en.wikipedia.org/wiki/Sylvester%27s_sequence
Hence our $A > B$ IF $a_1 = 2$.
I have no closed form for $B$, if you know one let me know. Digits of $B$ are nice too.
Analogue bounds exists for other values of $a_1$.
It seems related to integral transforms but in a discrete form.
Any ideas how to find $a_i$ for a given $A$ ?
In particular
$A = U + V \pi + W e$
Where $U,V,W$ are rationals and $\pi$ and $e$ are the famous constants
is of special interest to me.