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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)

%I M0843 #73 Apr 23 2023 15:41:52

%S 1,2,3,7,15,27,255,447,639,703,1819,4255,4591,9663,20895,26623,31911,

%T 60975,77671,113383,138367,159487,270271,665215,704511,1042431,

%U 1212415,1441407,1875711,1988859,2643183,2684647,3041127,3873535,4637979,5656191

%N In the '3x+1' problem, these values for the starting value set new records for highest point of trajectory before reaching 1.

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

%C Where records occur in A025586: A006885(n) = A025586(a(n)) and A025586(m) < A006885(n) for m < a(n). - _Reinhard Zumkeller_, May 11 2013

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

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

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

%H David Barina, <a href="/A006884/b006884.txt">Table of n, a(n) for n = 1..94</a> (first 89 terms from T. D. Noe and N. J. A. Sloane)

%H David Barina, <a href="http://pcbarina.fit.vutbr.cz/path-records.htm">Path records</a>.

%H Brian Hayes, <a href="https://www.jstor.org/stable/24969271">Computer Recreations: On the ups and downs of hailstone numbers</a>, Scientific American, 250 (No. 1, 1984), pp. 10-16.

%H J. C. Lagarias, <a href="http://www.cecm.sfu.ca/organics/papers/lagarias/paper/html/paper.html">The 3x+1 problem and its generalizations</a>, Amer. Math. Monthly, 92 (1985), 3-23.

%H G. T. Leavens and M. Vermeulen, <a href="http://dx.doi.org/10.1016/0898-1221(92)90034-F">3x+1 search problems</a>, Computers and Mathematics with Applications, 24 (1992), 79-99.

%H G. T. Leavens and M. Vermeulen, <a href="/A006877/a006877_1.pdf">3x+1 search programs</a>, Computers and Mathematics with Applications, 24 (1992), 79-99. (Annotated scanned copy)

%H Tomás Oliveira e Silva, <a href="http://sweet.ua.pt/tos/3x_plus_1.html">Tables</a> (gives many more terms)

%H Eric Roosendaal, <a href="http://www.ericr.nl/wondrous/pathrecs.html">3x+1 Path Records</a>

%H Robert G. Wilson v, <a href="/A006877/a006877.pdf">Letter to N. J. A. Sloane with attachments, Jan. 1989</a>

%H Robert G. Wilson v, <a href="/A006884/a006884.pdf">Tables of A6877, A6884, A6885, Jan. 1989</a>

%H <a href="/index/3#3x1">Index entries for sequences related to 3x+1 (or Collatz) problem</a>

%H <a href="/index/Go#GEB">Index entries for sequences from "Goedel, Escher, Bach"</a>

%t 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 *)

%t 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 *)

%o (Haskell)

%o a006884 n = a006884_list !! (n-1)

%o a006884_list = f 1 0 a025586_list where

%o f i r (x:xs) = if x > r then i : f (i + 1) x xs else f (i + 1) r xs

%o -- _Reinhard Zumkeller_, May 11 2013

%o (PARI) A025586(n)=my(r=n); while(n>2, if(n%2, n=3*n+1; if(n>r, r=n)); n>>=1); r

%o r=0; for(n=1,1e6, t=A025586(n); if(t>r, r=t; print1(n", "))) \\ _Charles R Greathouse IV_, May 25 2016

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

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

%K nonn,nice

%O 1,2

%A _N. J. A. Sloane_, _Robert Munafo_

%E b-file extended by _N. J. A. Sloane_, Nov 27 2007

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Last modified June 6 12:57 EDT 2023. Contains 363142 sequences. (Running on oeis4.)