Krdytkyu

I am looking for a value $a approx 14$ with some nice property. So I am going to define some things with this value $a$ and then ask what $a$ does the trick I want (If there is some $a$ that does the trick at all).

Definitions and Intro

Let $f(x,t) = frac{ln(t+a)^x}{t}$ and note that $f_x(x,t)=frac{ln(t+a)^{x}ln(ln(t+a))}{t}$ where $f_x$ refers to $frac{d}{dx} f(x,t)$

Now define $$g(x) = lim_{mtoinfty} sum_{t=1}^m f(x,t)-int_1^m f(x,t)dt $$

Note that $g(0) =gamma$ the Euler Mascheroni constant and the generalization above can be found under the generalization section of that wiki (So I am not conjuring this idea from thin air). In fact, when $a=0$ it seems that $g(x)$ is connected with what is referred to as Stieljes Constants.

It looks to me that there may exist some $a$ value that $g(x)=g'(x)$. Which would be kind of interesting. Because this would mean that $g(x)= gamma e^x$.

Here's a graph which led me to these suspicions. I won't reproduce the image of the graph because it just looks like $y=gamma e^x$. The interesting thing is that the numerical derivative nearly overlays the function.

The Question
Does there exist some $a$ that does this? And what is it?

Some preliminary notes/ attempts to make progress

We should note that $$g'(x) = lim_{mtoinfty} sum_{t=1}^m f_x(x,t)-int_1^m f_x(x,t)dt $$

Which allows for a little algebraic manipulations after we take the assumption $g'(x) =g(x)$. These manipulations haven't really helped me find out what $a$ is...

Motivations
David Hilbert referred to the puzzle of proving the irrationality of $gamma$ as "unapproachable." Which explains my title... I am just looking for some approaches to $gamma$ which may communicate some information about this constant.