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A255326
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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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2
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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
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,5
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COMMENTS
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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.
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LINKS
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FORMULA
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a(0) = 0; a(n) = 1 + a(A066503(n)).
Other identities:
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EXAMPLE
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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.
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MATHEMATICA
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a[n_] := -1 + Length@ NestWhileList[# - Times @@ FactorInteger[#][[;; , 1]] &, n, # > 0 &]; Array[a, 100, 0] (* Amiram Eldar, Mar 04 2024 *)
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PROG
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(Scheme, with memoization-macro definec from Antti Karttunen's IntSeq-library)
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CROSSREFS
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Cf. A255409 (gives the positions of records, also the first positions where a(n) = n).
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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