Could it be possible that the mass of the proton can be calculated by a series of integer sequences? Or is this just a curiosity?
Edit September 18, 2019 --- The most recent mass of the proton has diverged from this summation. It's a curiosity!
$$\sum_{m=1}^{\infty } \frac{1}{(m^2+1)_{2m}}=$$
NSum[1/Pochhammer[m^2+1,2m], {m,1,\[Infinity]}, WorkingPrecision -> 50]
Edit first eight digits match as of 2016. Question at math.SE
First seven digits match the proton's mass in kilograms.
$1.6726218229590580987863882056891582636342622102204\times10^{-27}$
$1.672621\times10^{-27}$ - from OEIS revised 11/15/12
$1.672621777\times10^{-27}$ - from Wikipedia
What's to say that sometime in the future, the proton's mass won't be made more accurate by adding $4.5\times10^{-35}$ to the current number?
Edit to explain motivation
Whenever I get a result I don't recognize, I look it up on OEIS. I found this number.
I posted on Mathematica.SE with the intention of asking for advice on how to prove that it converges. That would make this number a constant.
If this is a "fluke" or the result of "small numbers," it's still worth exploring.
Edit: It does converge.
Final Thoughts
$f_{p}=0.16726218229590580987863882056891582636342622102204$
is the 0-dimensional value of a fractal know as the Hilbert Curve.
To get the minimal 3-dimensional value: $f \times 10^{((dimension+1)!)}$ where $0\le dimension \le 3$.
This results in the value for a $1\times 1\times 1$ cube (coincidentally, the definition of the gram.)
To get kilograms: $f \times 10^{((dimension+1)!+3)}$.
I posit that the fractalness is the stabilizing influence on the proton.
Coda
I agree with everyone that I have been wrong-headed about the importance of this constant. I have posted the constant on OEIS A219733. Thanks for your patience.
Answer
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.
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