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Search: id:A078372
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| A078372 |
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Number of square-free integers in {n, f(n), f(f(n)), ...., 1}, where f is the Collatz function defined by f(x) = x/2 if x is even; f(x) = 3x + 1 if x is odd. |
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+0
2
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| 1, 2, 5, 2, 3, 6, 11, 2, 12, 4, 9, 6, 5, 12, 11, 2, 7, 12, 13, 4, 3, 10, 9, 6, 14, 6, 74, 12, 11, 12, 71, 2, 15, 8, 7, 12, 13, 14, 23, 4, 73, 4, 17, 10, 8, 10, 69, 6, 14, 14, 15, 6, 5, 74, 73, 12, 19, 12, 21, 12, 11, 72, 72, 2, 15, 16, 17, 8, 7, 8, 67, 12, 75, 14, 6, 14, 13, 24, 23, 4
(list; graph; listen)
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OFFSET
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1,2
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COMMENT
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Number of square-free terms in 3x+1 trajectory started at n.
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LINKS
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J. C. Lagarias, The 3x+1 problem and its generalizations, Amer. Math. Monthly, 92 (1985), 3-23.
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EXAMPLE
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The finite sequence n, f(n), f(f(n)), ...., 1 for n = 12 is 12, 6, 3, 10, 5, 16, 8, 4, 2, 1, which has six square-free terms. Hence a(12) = 6.
n=61: trajectory={61,184,92,46,23,70,35,...,20,10,5,16,8,4,2,1}, square-free terms={61,46,23,70,35,106,53,10,5,2,1}, so a(61)=11.
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MATHEMATICA
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<< NumberTheory`NumberTheoryFunctions` f[n_] := n/2 /; Mod[n, 2] == 0 f[n_] := 3 n + 1 /; Mod[n, 2] == 1 s[n_] := Module[{i, j}, i = n; j = 1; While[i > 1, If[SquareFreeQ[i], j = j + 1]; i = f[i]]; j]; Table[s[n], {n, 1, 100}]
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CROSSREFS
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Adjacent sequences: A078369 A078370 A078371 this_sequence A078373 A078374 A078375
Sequence in context: A132743 A111232 A087892 this_sequence A083381 A129396 A085220
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KEYWORD
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nonn
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AUTHOR
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Joseph L. Pe (joseph_l_pe(AT)hotmail.com), Dec 24 2002
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