OFFSET
1,4
COMMENTS
The fixed points of the x -> A181819(x) map are 1 and 2. Note that the x -> A000005(x) map has the same fixed points, and that A000005(n) = A181819(n) iff n is cubefree (cf. A004709). Under the x -> A181819(x) map, it seems significantly easier to generalize about which kinds of integers take a given number of iterations to reach a fixed point than under the x -> A000005(x) map.
Also the number of steps in the reduction of the multiset of prime factors of n wherein one repeatedly takes the multiset of multiplicities. For example, the a(90) = 5 steps are {2,3,3,5} -> {1,1,2} -> {1,2} -> {1,1} -> {2} -> {1}. - Gus Wiseman, May 13 2018
LINKS
Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Fixed Point
Eric Weisstein's World of Mathematics, Map
FORMULA
For n > 2, a(n) = a(A181819(n)) + 1.
a(n) = 0 iff n equals 1 or 2.
a(n) = 1 iff n is an odd prime (A000040(n) for n>1).
a(n) = 2 iff n is a composite perfect prime power (A025475(n) for n>1).
a(n) = 3 iff n is a squarefree composite integer or a power of a squarefree composite integer (cf. A182853).
a(n) = 4 iff n's prime signature a) contains at least two distinct numbers, and b) contains no number that occurs less often than any other number (cf. A182854).
EXAMPLE
MATHEMATICA
Table[If[n<=2, 0, Length[FixedPointList[Sort[Length/@Split[#]]&, Sort[Last/@FactorInteger[n]]]]-1], {n, 100}] (* Gus Wiseman, May 13 2018 *)
PROG
(Haskell)
a182850 n = length $ takeWhile (`notElem` [1, 2]) $ iterate a181819 n
-- Reinhard Zumkeller, Mar 26 2012
(Scheme, with memoization-macro definec)
CROSSREFS
KEYWORD
nonn
AUTHOR
Matthew Vandermast, Jan 04 2011
STATUS
approved