[seqfan] The Kaprekar 10% Cascade and A Question

[Hans Havermann]

When T.D. Noe included his Mathematica formulation for Kaprekar  
numbers (A006886) two months ago, I complained to him that it did not  
correctly determine *all* numbers up to the k-value inserted into the  
program. The reason apparently being that, beginning with k=4, k+1, k 
+2, etc. can contribute to the k-digit Kaprekar count. For example,  
while k=4 provides *seven* 4-digit Kaprekars, there are in fact *nine*  
4-digit Kaprekars, one provided by the heuristic's k=8 and another by  
k=9. Tony subsequently altered the heuristic to take better account of  
this (lest someone else fall into my mistaken assumption).

The provision of smaller-than-k-digit Kaprekars for a given k appears  
to follow a to-me-surprising (slightly probabilistic) 10%-cascade  
rule. For example, using k=280, the heuristic determines:

280-digit Kaprekars: 3865471147
279-digit Kaprekars:  386543753
278-digit Kaprekars:   38659566
277-digit Kaprekars:    3863599
276-digit Kaprekars:     386224
275-digit Kaprekars:      38782
274-digit Kaprekars:       3805
273-digit Kaprekars:        367
272-digit Kaprekars:         44
271-digit Kaprekars:          5
270-digit Kaprekars:          1
269-digit Kaprekars:          2

There can be gaps. For example, using k=196:

196-digit Kaprekars: 14733
195-digit Kaprekars:  1484
194-digit Kaprekars:   147
193-digit Kaprekars:    18
188-digit Kaprekars:     1 (!)

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?

{1.5, 2., 2.66667, 1.71429, 1.71429, 2.06452, 1.71429, 1.54839,  
2.85714, 1.54839, 1.71429, 2.01575, 1.6, 1.54839, 2.03175, 1.51181,  
1.71429, 2.5098, 2., 1.50588, 2.01575, 2.26772, 2., 2.00196, 1.52381,  
1.51181, 3.09677, 1.50294, 1.52381, 2.00024, 1.6, 1.50037, 2.03175,  
1.51181, 1.50588, 2.50122, 1.6, 1.6, 2.03175, 1.50037, 1.54839,  
3.00018, 1.54839, 2.2511, 2.50489, 1.51181, 1.71429, 2.00024, 1.54839,  
1.50073, 2.00784, 1.50147, 1.54839, 3.00073, 1.50294, 1.50018,  
2.03175, 1.50294, 1.71429, 2., 1.50588, 1.52381, 2.50031, 1.50002,  
1.50588, 3.00018, 1.6, 1.50073, 2.03175, 1.50018, 1.71429, 2.50002,  
1.6, 1.50588, 2.00049, 1.51181, 1.50294, 3.00009, 1.51181, 1.50002,  
3.50342, 1.50588, 1.6, 3., 1.50588, 1.50294, 2.00196, 2.25007,  
1.52381, 2.5, 1.50018, 1.50073, 2.03175, 1.51181, 1.50294, 2., 1.6,  
1.50294, 2.50122, 1.50001, 1.6, 2.00003, 1.6, 1.50009, 2.00002,  
1.51181, 1.50294, 3.00001, 1.54839, 2.25001, 3.01176, 1.50001, 1.6,  
2.00049, 1.50294, 1.50009, 2.50122, 1.51181, 1.50294, 2., 1.51181,  
1.50073, 2.00196, 1.50294, 1.50147, 3.75, 1.6, 1.5, 2.00391, 1.50009,  
1.50147, 3., 1.51181, 1.52381, 3.00005, 1.50005, 1.52381, 2.00003,  
1.71429, 1.5, 2.01575, 1.52381, 1.50147, 2.5, 1.50037, 1.50588,  
2.00012, 1.50005, 1.6, 2., 1.52381, 1.50018, 2.50008, 2.25, 1.50294,  
3., 1.54839, 1.50073, 2.00196, 1.5, 1.50294, 3.5, 1.6, 1.50073,  
2.00002, 1.50073, 1.54839, 3., 1.51181, 1.50002, 2.50244, 1.50037,  
1.54839, 2.00001, 1.50009, 2.25, 2.01575, 1.50147, 1.52381, 2.5,  
1.51181, 1.5, 2.00049, 1.50005, 1.50147, 2.00012, 1.50588, 1.50073,  
3.00001, 1.50005, 1.6, 2., 1.50294, 1.50588, 2.00003, 1.50009,  
1.54839, 3.75, 1.54839, 1.5, 2.00391, 1.50147, 1.50037, 2., 2.2522,  
1.50037, 2.50244, 1.5, 1.50294}