A078372 Number of squarefree 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.

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

Offset

1,2

Comments

Number of squarefree terms in 3x+1 trajectory started at n.

Links

Harvey P. Dale, Table of n, a(n) for n = 1..1000

J. C. Lagarias, The 3x+1 problem and its generalizations, Amer. Math. Monthly, 92 (1985), 3-23.

Example

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 squarefree terms. Hence a(12) = 6.
n=61: trajectory={61,184,92,46,23,70,35,...,20,10,5,16,8,4,2,1}, squarefree terms={61,46,23,70,35,106,53,10,5,2,1}, so a(61)=11.

Mathematica

Table[Count[NestWhileList[If[EvenQ[#], #/2, 3#+1]&, n, #>1&], _?(SquareFreeQ)], {n, 80}] (* Harvey P. Dale, Oct 23 2011 *)

Crossrefs

Sequence in context: A199611 A111232 A087892 * A154751 A299777 A197545

Adjacent sequences: A078369 A078370 A078371 * A078373 A078374 A078375

Keyword

nonn

Author

Joseph L. Pe, Dec 24 2002

Status

approved