Number-Of-Iteration Sequences
Leroy Quet
qq-quet at mindspring.com
Thu Mar 18 01:38:01 CET 2004
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
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