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%I #17 Jul 04 2021 15:52:46
%S 1,1,1,2,1,2,1,2,2,2,1,2,1,2,2,2,1,2,1,2,2,2,1,2,2,2,2,2,1,3,1,3,2,2,
%T 2,3,1,2,2,2,1,3,1,2,2,2,1,2,2,2,2,2,1,2,2,2,2,2,1,3,1,2,2,2,2,3,1,2,
%U 2,3,1,3,1,2,2,2,2,3,1,2,2,2,1,3,2,2,2,2,1,3,2,2,2,2,2,3,1,2,2,3,1,3,1,2,3,2,1,3,1,3
%N a(n) is the number of iterations needed for n to reach 0 under the mapping x -> A055212(x).
%C Since x > A055212(x) for all positive integers x, and the smallest value of A055212(x) is 0, every trajectory under iteration of the mapping x -> A055212(x) will end at 0.
%C If n = 1 or n is prime, then 0 will be reached in just one iteration of the mapping. Moreover, a(1), a(2), and a(3) form the only run of three consecutive 1's. All other 1's are isolated according to the prime numbers greater than 3.
%C If n is a composite number, then its trajectory under the mapping consists of a first step n -> A055212(n) followed by a(A055212(n)) steps to reach 0. So, a(n) = a(A055212(n)) + 1.
%e a(1) = 1, since 1 -> 0.
%e a(p) = 1, since p -> 0 for any prime p.
%e a(4) = 2, since 4 -> 1 -> 0.
%e a(30) = 3, since 30 -> 4 -> 1 -> 0.
%e a(1440) = 4, since 1440 -> 32 -> 4 -> 1 -> 0.
%t f[0] = 0; f[n_] := DivisorSigma[0, n] - PrimeNu[n] - 1; a[n_] := -2 + Length @ FixedPointList[f, n]; Array[a, 100] (* _Amiram Eldar_, Jun 03 2021 *)
%Y Cf. A000005, A036459, A055212.
%K nonn
%O 1,4
%A _Timothy L. Tiffin_, May 26 2021
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