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A378423 a(n) is the number of distinct terms reached by iterating the function f(x) = 2 + A008472(x), starting from x=n.

%I A378423 #12 Dec 12 2024 23:16:25
%S A378423 3,2,4,1,3,4,3,2,3,4,7,4,6,8,5,2,7,4,6,4,5,6,5,4,4,8,4,8,5,5,4,2,3,6,
%T A378423 9,4,6,6,5,4,7,9,6,6,5,5,5,4,4,4,7,8,6,4,5,8,5,4,7,5,6,10,5,2,5,5,10,
%U A378423 6,9,3,7,4,6,8,5,6,5,5,5,4,4,6,6,9,5,6,7,6,8,5,7,5,5,8,9,4,4,8,3,4
%N A378423 a(n) is the number of distinct terms reached by iterating the function f(x) = 2 + A008472(x), starting from x=n.
%C A378423 a(n)= The number of distinct elements in the set A(n)={f^{k}(n);k>=0}, where f^{k} is the k-th iteration of f.
%C A378423 The set A(n) contains either the fixed point 4 or a cyclic component {5,7,9}.
%e A378423 For n=33, 33->16->4->4-> ... and 4 is a fixed point, then a(n)= number of distinct terms = 3.
%e A378423 For n=66, 66->18->7->9->5->7 ... and {5,7,9} is a cyclic component, then a(n)= number of distinct terms = 5.
%p A378423 f:= proc(n)
%p A378423 add( d, d= numtheory[factorset](n)):
%p A378423 end proc: f(1) := 0:
%p A378423 g:= proc(n)
%p A378423    2 + f(n)
%p A378423 end proc:
%p A378423  a:= proc(n)
%p A378423  local k, result:
%p A378423  k := 1:
%p A378423 result := n:
%p A378423 while not (result = 4 or result = 5 or result = 7 or result = 9) do
%p A378423 result := g(result):
%p A378423 k := k + 1:
%p A378423 end do:
%p A378423 if result = 5 or result = 7 or result = 9 then
%p A378423 return k + 2;
%p A378423 else
%p A378423 return k:
%p A378423 end if
%p A378423 end proc:
%p A378423 map(a, [$1..100]);
%t A378423 a[n_] := -1 + Length@ NestWhileList[2 + If[# == 1, 0, Total[FactorInteger[#][[;; , 1]]]] &, n, UnsameQ, All]; Array[a, 100] (* _Amiram Eldar_, Nov 26 2024 *)
%o A378423 (Python)
%o A378423 from sympy import factorint
%o A378423 def a(n):
%o A378423     reach = set()
%o A378423     while n not in reach:
%o A378423         reach.add(n)
%o A378423         n = 2 + sum(factorint(n))
%o A378423     return len(reach)
%o A378423 print([a(n) for n in range(1, 101)]) # _Michael S. Branicky_, Nov 26 2024
%Y A378423 Cf. A008472.
%K A378423 nonn
%O A378423 1,1
%A A378423 _Rafik Khalfi_, Nov 25 2024