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A182850 a(n) = number of iterations that n requires to reach a fixed point under the x -> A181819(x) map. 101
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 (list; graph; refs; listen; history; text; internal format)
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

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.

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)

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

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.

Sequence in context: A305818 A303757 A323014 * A293227 A291208 A241165

Adjacent sequences:  A182847 A182848 A182849 * A182851 A182852 A182853

KEYWORD

nonn

AUTHOR

Matthew Vandermast, Jan 04 2011

STATUS

approved

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Last modified August 3 11:57 EDT 2020. Contains 336198 sequences. (Running on oeis4.)