A182850 - OEIS

A182850

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

126

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

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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.