Maximum Entropy when skewing mean

A great article you wrote, Nicole.

I tried to find an intuitive and empirical way to solve your puzzle. I tried the following hypothesis:

Given a random variable X with values in [1,n], by multiplying max entropy probabilities (1/n) of each value k of X in by a factor a^k (and then normalizing to get the sum of probabilities to 1), one can skew the distribution to any mean value between 1 and n yet keeping entropy as high as possible... When a=1, there is no skewing. When a>1 the mean value is higher than 1/n (and vice versa). In the dice case n=6. It seems to work!

Maybe you are able to prove this approach equivalent to yours?

I use solver in Excel to find the skewing factor ‘a’ fitting the E(X) I target.

E.g. to achieve E(X)=4.7 I find a=1,589282