This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A182850 #25 Apr 15 2025 14:58:48
%S A182850 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,
%T A182850 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,
%U A182850 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
%N A182850 a(n) = number of iterations that n requires to reach a fixed point under the x -> A181819(x) map.
%C A182850 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.
%C A182850 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
%H A182850 Reinhard Zumkeller, <a href="/A182850/b182850.txt">Table of n, a(n) for n = 1..10000</a>
%H A182850 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/FixedPoint.html">Fixed Point</a>
%H A182850 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Map.html">Map</a>
%F A182850 For n > 2, a(n) = a(A181819(n)) + 1.
%F A182850 a(n) = 0 iff n equals 1 or 2.
%F A182850 a(n) = 1 iff n is an odd prime (A000040(n) for n>1).
%F A182850 a(n) = 2 iff n is a composite perfect prime power (A025475(n) for n>1).
%F A182850 a(n) = 3 iff n is a squarefree composite integer or a power of a squarefree composite integer (cf. A182853).
%F A182850 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).
%e A182850 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.
%t A182850 Table[If[n<=2,0,Length[FixedPointList[Sort[Length/@Split[#]]&,Sort[Last/@FactorInteger[n]]]]-1],{n,100}] (* _Gus Wiseman_, May 13 2018 *)
%o A182850 (Haskell)
%o A182850 a182850 n = length $ takeWhile (`notElem` [1,2]) $ iterate a181819 n
%o A182850 -- _Reinhard Zumkeller_, Mar 26 2012
%o A182850 (Scheme)
%o A182850 ;; With memoization-macro definec.
%o A182850 (definec (A182850 n) (if (<= n 2) 0 (+ 1 (A182850 (A181819 n))))) ;; _Antti Karttunen_, Feb 05 2016
%Y A182850 A182857 gives values of n where a(n) increases to a record.
%Y A182850 Cf. A000961, A001222, A003434, A005117, A007916, A036459, A112798, A130091, A181819, A182851-A182858, A238748, A304455, A304464, A304465.
%K A182850 nonn
%O A182850 1,4
%A A182850 _Matthew Vandermast_, Jan 04 2011