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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.

%I A006884 M0843 #110 May 06 2025 10:57:28
%S A006884 1,2,3,7,15,27,255,447,639,703,1819,4255,4591,9663,20895,26623,31911,
%T A006884 60975,77671,113383,138367,159487,270271,665215,704511,1042431,
%U A006884 1212415,1441407,1875711,1988859,2643183,2684647,3041127,3873535,4637979,5656191
%N A006884 In the '3x+1' problem, these values for the starting value set new records for highest point of trajectory before reaching 1.
%C A006884 Both the 3x+1 steps and the halving steps are counted.
%C A006884 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 A006884 R. B. Banks, Slicing Pizzas, Racing Turtles and Further Adventures in Applied Mathematics, Princeton Univ. Press, 1999. See p. 96.
%D A006884 D. R. Hofstadter, Goedel, Escher, Bach: an Eternal Golden Braid, Random House, 1980, p. 400.
%D A006884 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A006884 David Barina, <a href="/A006884/b006884.txt">Table of n, a(n) for n = 1..98</a> (terms 1..84 from T. D. Noe, terms 85..89 from N. J. A. Sloane).
%H A006884 David Barina, <a href="http://pcbarina.fit.vutbr.cz/path-records.htm">Path records</a>.
%H A006884 David Barina, <a href="https://doi.org/10.1007/s11227-025-07337-0">Improved verification limit for the convergence of the Collatz conjecture</a>, J. Supercomputing (2025) Vol. 81, Art. No. 810. See p. 12.
%H A006884 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 A006884 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 A006884 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 A006884 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 A006884 Tomás Oliveira e Silva, <a href="http://sweet.ua.pt/tos/3x_plus_1.html">Tables</a> (gives many more terms).
%H A006884 Eric Roosendaal, <a href="http://www.ericr.nl/wondrous/pathrecs.html">3x+1 Path Records</a>.
%H A006884 Olivier Rozier and Claude Terracol, <a href="https://arxiv.org/abs/2502.00948">Paradoxical behavior in Collatz sequences</a>, arXiv:2502.00948 [math.GM], 2025. See p. 15.
%H A006884 Robert G. Wilson v, <a href="/A006877/a006877.pdf">Letter to N. J. A. Sloane with attachments, Jan. 1989</a>.
%H A006884 Robert G. Wilson v, <a href="/A006884/a006884.pdf">Tables of A6877, A6884, A6885, Jan. 1989</a>.
%H A006884 <a href="/index/3#3x1">Index entries for sequences related to 3x+1 (or Collatz) problem</a>
%H A006884 <a href="/index/Go#GEB">Index entries for sequences from "Goedel, Escher, Bach"</a>
%t A006884 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 A006884 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 A006884 (Haskell)
%o A006884 a006884 n = a006884_list !! (n-1)
%o A006884 a006884_list = f 1 0 a025586_list where
%o A006884    f i r (x:xs) = if x > r then i : f (i + 1) x xs else f (i + 1) r xs
%o A006884 -- _Reinhard Zumkeller_, May 11 2013
%o A006884 (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 A006884 r=0; for(n=1,1e6, t=A025586(n); if(t>r, r=t; print1(n", "))) \\ _Charles R Greathouse IV_, May 25 2016
%Y A006884 A060409 gives associated "dropping times", A060410 the maximal values and A060411 the steps at which the maxima occur.
%Y A006884 Cf. A006885, A006877, A006878, A033492, A060412-A060415, A132348.
%K A006884 nonn,nice
%O A006884 1,2
%A A006884 _N. J. A. Sloane_, _Robert Munafo_