A182850 a(n) = number of iterations that n requires to reach a fixed point under the x -> A181819(x) map.

0, 0, 1, 2, 1, 3, 1, 2, 2, 3, 1, 4, 1, 3, 3, 2, 1, 4, 1, 4, 3, 3, 1, 4, 2, 3, 2, 4, 1, 3, 1, 2, 3, 3, 3, 3, 1, 3, 3, 4, 1, 3, 1, 4, 4, 3, 1, 4, 2, 4, 3, 4, 1, 4, 3, 4, 3, 3, 1, 5, 1, 3, 4, 2, 3, 3, 1, 4, 3, 3, 1, 4, 1, 3, 4, 4, 3, 3, 1, 4, 2, 3, 1, 5, 3, 3, 3, 4, 1, 5, 3, 4, 3, 3, 3, 4, 1, 4, 4, 3, 1, 3, 1, 4, 3

Offset: 1

Author

Matthew Vandermast, Jan 04 2011

Keywords

• nonn

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

Examples

			A181819(6) = 4; A181819(4) = 3; A181819(3) = 2; A181819(2) = 2. Therefore, a(6) = 3, a(4) = 2, a(3)= 1, and a(2) = 0.
		

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

Crossrefs

A182857 gives values of n where a(n) increases to a record.

Cf. A000961, A001222, A003434, A005117, A007916, A036459, A112798, A130091, A181819, A182851-A182858, A238748, A304455, A304464, A304465.

Programs

Haskell

a182850 n = length $ takeWhile (`notElem` [1,2]) $ iterate a181819 n
-- Reinhard Zumkeller, Mar 26 2012

Mathematica

Table[If[n<=2,0,Length[FixedPointList[Sort[Length/@Split[#]]&,Sort[Last/@FactorInteger[n]]]]-1],{n,100}] (* Gus Wiseman, May 13 2018 *)

Scheme

;; With memoization-macro definec.
(definec (A182850 n) (if (<= n 2) 0 (+ 1 (A182850 (A181819 n))))) ;; Antti Karttunen, Feb 05 2016

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).