The number 13.01 quintillion, or its reciprocal, is a major unexplained number and also a very useful one.
You can get it (to somewhat better precision than shown here) from the NIST website's value for Planck mass and proton mass simply by dividing one by the other. In conventional notation: 1.301 × 1019.
The proton mass is 1/(13.01 quintillion) Planck mass.
The proton Compton wavelength is 13.01 quintillion Planck lengths.
(Angular format is used here throughout.)
The proton energy and frequency equivalents are 1/(13.01 quintiillion) times the Planck energy and the Planck frequency, as would be expected.
In metric terms, all the corresponding numbers differ from one another: the mass in kilograms, the Compton wavelength in meters, the energy in joules (and electron volts) and frequency in Hertz.. The values of all these things in different metric units are tabulated and posted on web as distinct physical constants, although at a basic level to do so is redundant. Using Planck units they would just post a single number.
Unfortunately the physics community in general does not have a notation for the proton/Planck mass ratio—approxately the reciprocal of 13 × 1018—so for the sake of present discussion I will call this tiny number ZETA. Here is a relevant quote from Frank Wilczek's June 2001 Physics Today article:
<<Soon after Max Planck introduced his constant h-bar in the course of a phenomenological fit to the blackbody radiation spectrum, he pointed out the possibility of building a system of units based on the three fundamental constants h-bar, c, and GN. Indeed, from these three we can define a unit of mass (h-bar c/GN)1/2, a unit of length (h-bar GN/c3)1/2, and a unit of time (h-bar GN/c5)1/2—what we now call the Planck mass, length, and time, respectively. Planck's proposal for a system of units based on fundamental physical constants was, when it was made, formally correct but rather thinly rooted in fundamental physics. Over the course of the 20th century, however, his proposal became compelling. Now there are profound reasons to regard c as the fundamental unit of velocity and as the fundamental unit of action. In the special theory of relativity, there are symmetries relating space and time—and c serves as a conversion factor between the units in which space intervals and time intervals are measured. In quantum theory, the energy of a state is proportional to the frequency of its oscillations—and is the conversion factor. Thus c and h-bar appear directly as primary units of measurement in the basic laws of these two great theories. Finally, in general relativity theory, spacetime curvature is proportional to the density of energy—and GN (actually c4/GN ) is the conversion factor. If we accept that GN is a primary quantity, together with h-bar and c, then the enigma of [ZETA2] smallness looks quite different. We see that the question it poses is not, "Why is gravity so feeble?" but rather, "Why is the proton's mass so small?" For in natural (Planck) units, the strength of gravity simply is what it is, a primary quantity, while the proton's mass is the tiny number [ZETA].>>
http://www.physicstoday.org/pt/vol-54/iss-6/p12.html
I seem to recall that the Chandrasekhar mass limit (mass leading to supernova explosion) can be expressed in terms of the reciprocal of zeta and the Planck mass, for a typical case of a star that is half protons, as
pi/4 × (13.01 quintillion)2 planck masses.
It is usually expressed in solar masses but is easily calculated in Planck masses and doing so shows that this zeta number (which still has no generally accepted symbol) gets around.
Wilczek's article suggests a comparison with another major unexplained number, the fine structure constant.