A255409 -id:A255409 - cp's OEIS frontend

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.

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, 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, 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

Offset: 0

Antti Karttunen, Mar 23 2015

nonn

In other words, number of iterations needed to reach zero with map x <- A066503(x), when starting from n.

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.

			Largest squarefree number dividing 27 is 3, and 27 - 3 = 24.
Largest squarefree number dividing 24 is 6, and 24 - 6 = 18.
Largest squarefree number dividing 18 is 6, and 18 - 6 = 12.
Largest squarefree number dividing 12 is 6, and 12 - 6 = 6.
Largest squarefree number dividing 6 is 6, and 6 - 6 = 0.
Thus a(6) = 1, a(12) = 2, a(18) = 3, a(24) = 4 and a(27) = 5.

Antti Karttunen, Table of n, a(n) for n = 0..10000

Cf. A255409 (gives the positions of records, also the first positions where a(n) = n).

Cf. A005117, A007947, A066503.

a[n_] := -1 + Length@ NestWhileList[# - Times @@ FactorInteger[#][[;; , 1]] &, n, # > 0 &]; Array[a, 100, 0] (* Amiram Eldar, Mar 04 2024 *)

a(0) = 0; a(n) = 1 + a(A066503(n)).
Other identities:
a(k) = 1 iff k = A005117(n).

cp's OEIS Frontend

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.

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