Number-Of-Iteration Sequences

[Leroy Quet]

Consider the sequence where a(n) =

sum{k>=1} k*c(k),

if n = 
product{k>=1} p(k)^c(k),

each c = a nonnegative integer, and p(k) is the k_th prime.

{a(k)} is:
http://www.research.att.com/projects/OEIS?Anum=A056239

(0,1,2,2,3,3,4,3,4,4,5,4,6,5,5,4,7,...)

Now, define b(n) as the number of iterations needed to get

a(a(...a(n)..)) = 0.

(same as, for n>= 2, number of a's until 1.) 

In other words,

b(1) = 0; b(n) = b(a(n)) + 1:


0,1,2,2,3,3,3,3,3,3,4,3,4,4,4,3,4,...

(unless I goofed)


My choice of {a(j)} = A056239 was almost arbitrary.

For this b-sequence, it might be more natural to add one or subtract one 
from each term for whatever reason. But I leave these choices to anyone 
who decides to submit the sequence,  if anyone wants to submit it.


I originally had wondered about the number-of-iterations sequence when
{a(k)} was A001414.
(a sequence similar to A056239:
A001414: "Name: Integer log of n: sum of primes dividing n (with 
repetition).")

But this b-sequence is already in the EIS:

http://www.research.att.com/projects/OEIS?Anum=A002217


Generally, I wonder about interesting sequences formed from other 
sequences, where the original sequences consist of positive integer terms 
<= their indexes, and the derived sequences are the number of iterations 
needed to achieve a fixed-point or 0 or 1 or whatever specified value.

I am guessing that there are some interesting number-of-iteration 
sequences of this kind which are derived from EIS sequences.

thanks,
Leroy Quet