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

0,5

Comments

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.

Links

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

Formula

a(0) = 0; a(n) = 1 + a(A066503(n)).

Other identities:

a(k) = 1 iff k = A005117(n).

Example

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.

Mathematica

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

Prog

(Scheme, with memoization-macro definec from Antti Karttunen's IntSeq-library)
(definec (A255326 n) (if (zero? n) n (+ 1 (A255326 (A066503 n)))))

Crossrefs

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

Cf. A005117, A007947, A066503.

Sequence in context: A307848 A358260 A368978 * A085424 A088737 A318434

Adjacent sequences: A255323 A255324 A255325 * A255327 A255328 A255329

Keyword

nonn

Author

Antti Karttunen, Mar 23 2015

Status

approved