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%I #71 Mar 26 2024 09:16:58
%S 1,2,4,8,16,20,24,32,40,48,52,56,64,68,72,80,84,88,96,100,104,112,116,
%T 128,132,136,144,148,152,160,168,176,180,184,192,196,200,208,212,224,
%U 228,232,240,244,256,260,264,272,276,280,288,296,304,308,312,320,324
%N Numbers m that are the largest number in their Collatz (3x+1) trajectory.
%C Or, possible peak values in 3x+1 trajectories: 1,2 and m=16k+4,16k+8,16k but not for all k; those 4k numbers [like m=16k+12 and others] which cannot be such peaks are listed in A087252.
%C Possible values of A025586(m) in increasing order. See A275109 (number of times each value of a(n) occurs in A025586). - _Jaroslav Krizek_, Jul 17 2016
%H Reinhard Zumkeller, <a href="/A033496/b033496.txt">Table of n, a(n) for n = 1..10000</a> (first 2000 terms from T. D. Noe)
%H Hartmut F. W. Hoft, <a href="/A033496/a033496.pdf">initial Collatz fans</a>
%H <a href="/index/3#3x1">Index entries for sequences related to 3x+1 (or Collatz) problem</a>
%F A008908(a(n)) = A159999(a(n)). - _Reinhard Zumkeller_, May 04 2009
%F Max(A070165(a(n),k): k=1..A008908(a(n))) = A070165(a(n),1) = a(n). - _Reinhard Zumkeller_, Oct 22 2015
%e These peak values occur in 1, 3, 6, 8, 9, 11, 12, 13, 14, 15, 16, 17, 18, 21, 22, 27, 30, 39, 44, 71, 75, 1579 [3x+1]-iteration trajectories started with different initial values. This list most probably is incomplete.
%e From _Hartmut F. W. Hoft_, Jun 24 2016: (Start)
%e Let n be the maximum in some Collatz trajectory and let F(n), the initial fan of n, be the set of all initial values less than or equal to n whose Collatz trajectories lead to n as their maximum. Then the size of F(n) never equals 2, 4, 5, 7 or 10 (see the link).
%e Conjecture: Every number k > 10 occurs as the size of F(n) for some n.
%e Fans F(n) of size k, for all 10 < k < 355, exist for 4 <= n <= 50,000,000. The largest fan in this range, F(41163712), has size 7450.
%e (End)
%t Collatz[a0_Integer, maxits_:1000] := NestWhileList[If[EvenQ[ # ], #/2, 3# + 1] &, a0, Unequal[ #, 1, -1, -10, -34] &, 1, maxits]; (* Collatz[n] function definition by Eric Weisstein *)
%t Select[Range[324], Max[Collatz[#]] == # &] (* _T. D. Noe_, Feb 28 2013 *)
%o (Haskell)
%o a033496 n = a033496_list !! (n-1)
%o a033496_list = 1 : filter f [2, 4 ..] where
%o f x = x == maximum (takeWhile (/= 1) $ iterate a006370 x)
%o -- _Reinhard Zumkeller_, Oct 22 2015
%o (Magma) Set(Sort([Max([k eq 1 select n else IsOdd(Self(k-1)) and not IsOne(Self(k-1)) select 3*Self(k-1)+1 else Self(k-1) div 2: k in [1..5*n]]): n in [1..2^10] | Max([k eq 1 select n else IsOdd(Self(k-1)) and not IsOne(Self(k-1)) select 3*Self(k-1)+1 else Self(k-1) div 2: k in [1..5*n]]) le 2^10])) // _Jaroslav Krizek_, Jul 17 2016
%o (Python)
%o def a(n):
%o if n<2: return [1]
%o l=[n, ]
%o while True:
%o if n%2==0: n//=2
%o else: n = 3*n + 1
%o if n not in l:
%o l.append(n)
%o if n<2: break
%o else: break
%o return l
%o print([n for n in range(1, 501) if max(a(n)) == n]) # _Indranil Ghosh_, Apr 14 2017
%Y Cf. A025586, A087251-A087256, A015999, A275109.
%Y Cf. A006370, A008908, A070165.
%Y Cf. A095384 (contains a definition of Collatz[]).
%Y Cf. A105730, A233293.
%K nonn,nice,easy
%O 1,2
%A _Jeff Burch_
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