A006884 In the '3x+1' problem, these values for the starting value set new records for highest point of trajectory before reaching 1. (Formerly M0843)

1, 2, 3, 7, 15, 27, 255, 447, 639, 703, 1819, 4255, 4591, 9663, 20895, 26623, 31911, 60975, 77671, 113383, 138367, 159487, 270271, 665215, 704511, 1042431, 1212415, 1441407, 1875711, 1988859, 2643183, 2684647, 3041127, 3873535, 4637979, 5656191

Offset

1,2

Comments

Both the 3x+1 steps and the halving steps are counted.

Where records occur in A025586: A006885(n) = A025586(a(n)) and A025586(m) < A006885(n) for m < a(n). - Reinhard Zumkeller, May 11 2013

References

R. B. Banks, Slicing Pizzas, Racing Turtles and Further Adventures in Applied Mathematics, Princeton Univ. Press, 1999. See p. 96.

D. R. Hofstadter, Goedel, Escher, Bach: an Eternal Golden Braid, Random House, 1980, p. 400.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

David Barina, Table of n, a(n) for n = 1..97 (terms 1..84 from T. D. Noe, terms 85..89 from N. J. A. Sloane)

David Barina, Path records.

Brian Hayes, Computer Recreations: On the ups and downs of hailstone numbers, Scientific American, 250 (No. 1, 1984), pp. 10-16.

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

G. T. Leavens and M. Vermeulen, 3x+1 search problems, Computers and Mathematics with Applications, 24 (1992), 79-99.

G. T. Leavens and M. Vermeulen, 3x+1 search programs, Computers and Mathematics with Applications, 24 (1992), 79-99. (Annotated scanned copy)

Tomás Oliveira e Silva, Tables (gives many more terms).

Eric Roosendaal, 3x+1 Path Records.

Robert G. Wilson v, Letter to N. J. A. Sloane with attachments, Jan. 1989.

Robert G. Wilson v, Tables of A6877, A6884, A6885, Jan. 1989.

Index entries for sequences related to 3x+1 (or Collatz) problem

Index entries for sequences from "Goedel, Escher, Bach"

Mathematica

mcoll[n_]:=Max@@NestWhileList[If[EvenQ[#], #/2, 3#+1]&, n, #>1&]; t={1, max=2}; Do[If[(y=mcoll[n])>max, max=y; AppendTo[t, n]], {n, 3, 705000, 4}]; t (* Jayanta Basu, May 28 2013 *)
DeleteDuplicates[Parallelize[Table[{n, Max[NestWhileList[If[EvenQ[#], #/2, 3#+1]&, n, #>1&]]}, {n, 57*10^5}]], GreaterEqual[#1[[2]], #2[[2]]]&][[;; , 1]] (* Harvey P. Dale, Apr 23 2023 *)

Prog

(Haskell)
a006884 n = a006884_list !! (n-1)
a006884_list = f 1 0 a025586_list where
   f i r (x:xs) = if x > r then i : f (i + 1) x xs else f (i + 1) r xs
-- Reinhard Zumkeller, May 11 2013
(PARI) A025586(n)=my(r=n); while(n>2, if(n%2, n=3*n+1; if(n>r, r=n)); n>>=1); r
r=0; for(n=1, 1e6, t=A025586(n); if(t>r, r=t; print1(n", "))) \\ Charles R Greathouse IV, May 25 2016

Crossrefs

A060409 gives associated "dropping times", A060410 the maximal values and A060411 the steps at which the maxima occur.

Cf. A006885, A006877, A006878, A033492, A060412-A060415, A132348.

Sequence in context: A209658 A098763 A001276 * A074742 A020873 A049958

Adjacent sequences: A006881 A006882 A006883 * A006885 A006886 A006887

Keyword

nonn,nice

Author

N. J. A. Sloane, Robert Munafo

Status

approved