[seqfan] Re: The Kaprekar 10% Cascade and A Question

[Hans Havermann]

> Let me compare the ratios of the number of divisors of 10^k-1  
> (A070528) with the total number of Kaprekars derived by using k in  
> the heuristic. For k=196, there are 24576 divisors of 10^196-1 and  
> 16383 (14733+1484+147+18+1) Kaprekars: the ratio ~1.5. I'm ending  
> with a list of divisors/Kaprekars ratios for 1 to 209. They appear  
> to cluster around some small rationals. Can anyone figure out what  
> it is that determines this ratio?

I may have figured out he broad brush-strokes myself: It seems to  
depend on the powers contained in the 10^k-1 factorization. Of course  
there will be some small-number artifacts (exceptions) to this but, in  
general, I think the ratio of 1.5 is the norm (the factorization of  
10^k-1 begins with 3^2). We get a ratio of 2.0 when the factorization  
of 10^k-1 begins with 3^3; 2.25, when it begins with 3^2 and contains  
some other n^2; 2.5, when it begins with 3^4; 3.0, when it begins with  
3^5, or when it begins with 3^3 and contains some other n^2; 3.5, when  
it begins with 3^6; 3.75, when it begins with 3^4 and contains some  
other n^2. This will prove helpful in estimating the 10%-cascade for  
the five k less than 280 for which I am currently unable to calculate  
actual numbers because of the large number of divisors of 10^k-1.