Have I discovered how to calculate the proton's mass using only integers?
To formalize dushya's comment as an answer:
Since the kilogram is an arbitrary, man-made unit, the actual numerical value of the proton mass in kilograms is meaningless (i.e. it's as good as its value in pounds, ounces, stones, solar masses, $\textrm{MT}/c^2$, etc.). The true fundamental constants of nature are dimensionless: they have the same value in every unit system. Thus dimensional constants like $c$, $\hbar$, $G$, and indeed $m_p$ and $m_e$, are not very meaningful and can be set to $1$ with a judicious choice of units (which is done quite often).
True fundamental constants are often ratios of dimensional quantities such as the fine structure constant, $$\alpha=\frac{e^2/4\pi\epsilon_0}{\hbar c},$$ which quantifies how strong, on a quantum scale, the electromagnetic interaction is. In terms of mass, the constants you'd like to predict are things like the ratio $m_p/m_e\approx 1800$, and so on.
Given that, the formula you have found is just a fluke: a consequence of the fact that we chose as our basic unit of mass the mass of a cube of water whose sides measure one hundred-millionth of a quarter of a meridian.
EDIT, given the long comment thread:
@Fred, let me try and rephrase this a bit to see if I can bring out the arbitrariness we're talking about well up to the surface. The real number you have discovered is the inverse of the one you posted: $$\frac{10^{26}}{\sum_{m=1}^\infty \frac{1}{(m^2+1)_{2m}}}\approx 5.978638 \times 10^{26},$$ which appears to approximate within experimental error the number of protons and neutrons that will fit - at sea level and at "room" temperature - a cubical box about yea big in side containing that particular common chemical that you find in drinking fountains, kitchen sinks, lakes, and even falling out of the sky (on Earth) rather often.
Since the proton really is quite fundamental, any stabilizing influence of the fractalness needs to account for the size of the Earth, its predominant climate a hundred years ago, the abundance of water in it, and the detailed chemical state of the brains of a number of mainly French gentlemen that sat down a while ago to try and make unit systems (which are always arbitrary) at least simple to work with.
This does allow you to make a prediction--- the significance of the match tells you approximately the number of attempts you have made to get such a coincidence. The ratio of the mass of a proton to a mass of a steel cylinder in Paris was determined by the psychology of some French revolutionaries.
But from the accuracy you get, one can be 99.99% sure that you automatically went through an automated search of hundreds of thousands of integer sequences, probably by going through the database, and compared to a list of about 100 physical constants. When one is doing such a search, one should say the number of attempts.
A historical case where something like this was significant was when people in the early part of the 19th century noticed that ${1\over \sqrt{\epsilon_0 \mu_0}}= c$. This came as a surprise back then (the system of units didn't make it a definition as it does today, nor was it obvious from the relativity postulate). There aren't many more cases of numerology like this being significant.
no, I am afraid you have not discovered a physically relevant relation until you prove what is the relation between the series and the terms defining the theory in the UV (such as $g_{3}$ at some scale $\mu$, $m_{u,d}$, electromagnetic corrections...). I would be much more impressed if you could get at the same time a similar formula for the neutron mass which is 'almost' like a proton beside the small up- and donw-quark masses and the electric charge. As I said, these things should enter in your formula in order to give you $m_{proton}\neq m_{neutron}$.