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A255326 a(n) gives the number of steps needed to reach zero, when we start from x = n and repeatedly subtract x's squarefree kernel (A007947(x)) from it.

%I A255326 #21 Mar 04 2024 01:38:07
%S A255326 0,1,1,1,2,1,1,1,2,2,1,1,2,1,1,1,2,1,3,1,2,1,1,1,4,3,1,5,2,1,1,1,2,1,
%T A255326 1,1,2,1,1,1,2,1,1,1,2,2,1,1,2,2,3,1,2,1,3,1,2,1,1,1,2,1,1,2,2,1,1,1,
%U A255326 2,1,1,1,2,1,1,3,2,1,1,1,2,2,1,1,2,1,1,1,2,1,3,1,2,1,1,1,4,1,3,2,4,1,1,1,2,1,1,1,2,1,1,1,4,1,1,1,2,2,1,1,4
%N A255326 a(n) gives the number of steps needed to reach zero, when we start from x = n and repeatedly subtract x's squarefree kernel (A007947(x)) from it.
%C A255326 In other words, number of iterations needed to reach zero with map x <- A066503(x), when starting from n.
%C A255326 Also, for n >= 1, a(n) = one more than the number of steps to reach a squarefree number (A005117) when we repeatedly subtract the largest squarefree number dividing x, starting from x <- n.
%H A255326 Antti Karttunen, <a href="/A255326/b255326.txt">Table of n, a(n) for n = 0..10000</a>
%F A255326 a(0) = 0; a(n) = 1 + a(A066503(n)).
%F A255326 Other identities:
%F A255326 a(k) = 1 iff k = A005117(n).
%e A255326 Largest squarefree number dividing 27 is 3, and 27 - 3 = 24.
%e A255326 Largest squarefree number dividing 24 is 6, and 24 - 6 = 18.
%e A255326 Largest squarefree number dividing 18 is 6, and 18 - 6 = 12.
%e A255326 Largest squarefree number dividing 12 is 6, and 12 - 6 = 6.
%e A255326 Largest squarefree number dividing 6 is 6, and 6 - 6 = 0.
%e A255326 Thus a(6) = 1, a(12) = 2, a(18) = 3, a(24) = 4 and a(27) = 5.
%t A255326 a[n_] := -1 + Length@ NestWhileList[# - Times @@ FactorInteger[#][[;; , 1]] &, n, # > 0 &]; Array[a, 100, 0] (* _Amiram Eldar_, Mar 04 2024 *)
%o A255326 (Scheme, with memoization-macro definec from _Antti Karttunen_'s IntSeq-library)
%o A255326 (definec (A255326 n) (if (zero? n) n (+ 1 (A255326 (A066503 n)))))
%Y A255326 Cf. A255409 (gives the positions of records, also the first positions where a(n) = n).
%Y A255326 Cf. A005117, A007947, A066503.
%K A255326 nonn
%O A255326 0,5
%A A255326 _Antti Karttunen_, Mar 23 2015