A006885 Record highest point of trajectory before reaching 1 in '3x+1' problem, corresponding to starting values in A006884.

1, 2, 16, 52, 160, 9232, 13120, 39364, 41524, 250504, 1276936, 6810136, 8153620, 27114424, 50143264, 106358020, 121012864, 593279152, 1570824736, 2482111348, 2798323360, 17202377752, 24648077896, 52483285312, 56991483520, 90239155648, 139646736808

Offset: 1

Author

Robert Munafo

Keywords

• nonn
• nice

Comments

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

Record values in A025586: a(n) = A025586(A006884(n)) and A025586(m) < a(n) for m < A006884(n). - Reinhard Zumkeller, May 11 2013

In an email of Aug 06 2023, Guy Chouraqui observes that the digital root of a(n) appears to be 7 for all n > 2. - N. J. A. Sloane, Aug 11 2023

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.
• G. T. Leavens and M. Vermeulen, 3x+1 search problems, Computers and Mathematics with Applications, 24 (1992), 79-99.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Hugo Pfoertner, Table of n, a(n) for n = 1..97, using data from Eric Rosendaal's 3x+1 Path records (terms 1..84 from T. D. Noe)
• 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 programs, Computers and Mathematics with Applications, 24 (1992), 79-99. (Annotated scanned copy)
• Peter Munn, Plot2 graph of record highest point against starting value
• 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 from "Goedel, Escher, Bach"
• Index entries for sequences related to 3x+1 (or Collatz) problem

Crossrefs

Cf. A006884, A006877, A006878, A033492.

Programs

Haskell

a006885 = a025586 . a006884  -- Reinhard Zumkeller, May 11 2013

Mathematica

mcoll[n_]:=Max@@NestWhileList[If[EvenQ[#],#/2,3#+1]&,n,#>=n&]; t={1,max=2}; Do[If[(y=mcoll[n])>max,AppendTo[t,max=y]],{n,3,10^6,4}]; t (* Jayanta Basu, May 28 2013 *)