[seqfan] Iterated Product of Alternating Digits

[Paul D Hanna]

     Here are some interesting sequences posed by Amarnath Murthy.
 
Below, Amarnath's function is defined along with the initial terms
of four sequences generated by the iterations of this function.
 
My main concern is the sequence Seq3 that I describe below.
The final term that I provide, a(25)=2771717, 
is the last term that I calculate under 15,000,000.
That is, I do not find a number between n=2771717 and 15million
that requires more than 25 iterations of f(n) to reach a single digit.
 
Is this correct?  Could someone verify this, or find the 
next number that requires more than 25 iterations of f(n) 
to reach a single digit?
 
    Thanks,
        Paul D Hanna
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Define a function f(n) of a positive integer n 
as the product of the two numbers formed from 
the alternating digits of n:
e.g.  f(1234) = 13*24 = 312.
e.g.  f(12345) = 135*24 = 3240.
 
Some sequences generated by f(n) are as follows.
 
Seq1. The final result of iterating f(n) until a 
      single digit number is reached.
e.g.  f(1234)=312, f(312)=32*1=32, f(32)=3*2=6.
 
1,2,3,4,5,6,7,8,9,0,1,2,3,4,5,6,7,8,9,0,2,4,6,8,0,
2,4,6,8,0,3,6,9,2,5,8,2,8,4,0,4,8,2,6,0,8,6,6,8,0,
5,0,5,0,0,0,5,0,0,0,6,2,8,8,0,8,8,6,0,0,7,4,2,6,5,
8,8,0,8,0,8,6,8,6,0,6,0,8,4,0,9,8,4,8,0,0,8,4,8,0,
0,0,0,0,0,0,0,0,0,0,1,2,3,4,5,6,7,8,9,0,4,8,2,6,0,
6,2,8,8,0,9,8,4,8,0,6,5,0,5,0,6,6,0,0,0,8,6,4,8,0,...
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Seq2. The number of iterations of f(n) needed to reach 
      a single digit number.
0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,
2,2,2,2,1,1,1,1,2,2,2,2,2,3,1,1,1,2,2,2,2,3,2,3,1,
1,2,2,2,3,2,3,2,3,1,1,2,2,2,2,3,2,3,3,1,1,2,2,3,3,
2,4,3,3,1,1,2,2,2,2,3,3,3,3,1,1,2,3,3,3,3,3,3,2,1,
1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,2,
2,3,3,3,2,2,3,4,2,3,3,2,3,4,2,3,3,3,3,2,3,4,3,3,2,...
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Seq3. The smallest numbers a(n) that require n iterations 
      of f(k) to reach a single digit, with a(0)=1.
 
1,10,25,39,77,171,199,577,887,1592,2682,3988,6913,
18747,39577,58439,99428,173442,267343,299137,574182,
685812,880543,1635812,1974447,2771717,...

The next term appears to be greater than 15,000,000.
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Seq4. Iterations of the inverse of f(k) such that
      the sequence is infinite, starting with 6.

6,32,48,224,288,392,576,966,2432,6348,6992,13884,
33596,47544,58646,87216,99428,173442,281212,390176,
574182,863876,986928,...

Even though there does not exist a unique 'inverse' 
to f(k), these terms of Seq4 seem to be the 
reverse order of the limit (if one exists) 
of iterations of f(k) on the terms in Seq3.

Examples of the iterations of f(k) on the terms of Seq3 are:

880543,685812,574182,390176,281212,173442,99428,
       87216,58646,47544,33596,13884,6992,6348,
       2432,966,576,392,288,224,48,32,6
 
1635812,899682,863876,574182,390176,281212,173442,
       99428,87216,58646,47544,33596,13884,6992,
       6348,2432,966,576,392,288,224,48,32,6
 
1974447,1649168,986928,863876,574182,390176,281212,
       173442,99428,87216,58646,47544,33596,13884,
       6992,6348,2432,966,576,392,288,224,48,32,6
 
2771717,1974447,1649168,986928,863876,574182,390176,
       281212,173442,99428,87216,58646,47544,33596,
       13884,6992,6348,2432,966,576,392,288,224,48,32,6

Do the successive iterations of f(k) on the 
rest of the terms of Seq3 tend to have these 
same trailing terms:
{...,6992,6348,2432,966,576,392,288,224,48,32,6}?

Are these limiting terms (if limit exists) always even?
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