Estimates of the number of magic squares, cubes, ... (hypercubes)

The Enumber can be calculated exactly or at least with high accuracy.
But the Enumber of certain magic squares (cubes, ...) is not exactly equal to their total number.
The aberration shall be described by a correction factor: the Cfactor
Definition
Consider magic hypercubes (squares, cubes, ...) of order m and dimension n with certain magic properties.
Say N is the total number of these magic objects and E their Enumber.
Then the correctionfactor C is defined as C = N / E .
So we can write:
N = C · E  Number of magic squares, cubes, ... 
For magic squares and semimagic squares of orders up to 20 we already have got approximations
by Monte Carlo methods with a relative error smaller than 0.5%.
This enables us to approximate some Cfactors.
The Cfactor increases constantly up to order 17 and can be approximated by the linear function C_{~}(m):
C(m) ≈ C_{~}(m) = 0.185 · (m  1)  Cfactor approximation for magic squares with m ≥ 6 
For semimagic squares we may assume:
C_{s}(m) ≈ C_{s~}(m) = 0.1805 · (m + 1.5)  Cfactor approximation for semimagic squares with m ≥ 6 
Note that the magicseriesmethod (Enumber) does not really work for orders smaller than 6.
There are too much symmetries and special properties of these magic squares.
Thus there are more magic squares than we would expect by our statistics.
The following diagram shows the Cfactors for magic squares (red) and semimagic squares (yellow) together with their linear approximation functions.
Click on the diagram in order to enlarge it.
The nearly linear increase of the Cfactor for orders <18 is an experimental result.
I haven't got a good analytical explanation.
Probably the Cfactor increases slower for orders > 17.
Further approximations with Monte Carlo methods would be necessary.
For orders greater than 20 we may assume that N(m) is greater than E(m) and smaller than C_{~}(m) · E(m).
Consider C_{~}(m) · E(m) as a slightly more accurate estimate for N(m) than E(m).
For magic squares of even orders the Cfactor is slightly above the blue line and for odd orders below this line.
The probability to be magic is lower for diagonals of odd orders because they share the number in the center.
The following diagram shows the differences between the Cfactors C(m) and their linear approximations C_{~}(m).
Click on the diagram in order to enlarge it.
I can't explain why the Cfactors for order 18 and 19 are too small.
This would mean that the diagonals must have a lower probability of being magic, because the number of semimagic squares increases constantly.
Probably there is a bug in one of my approximation programs.
Or does anybody know why there are less magic squares of order 18 and 19 than expected?
Walter Trump, Nürnberg, Germany, (c) 20050221 (last modified: 20051029)