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%I #16 Aug 12 2014 04:05:11
%S 27,31,41,54,47,55,62,63,73,71,82,83,97,94,9,95,7,108,3,109,11,14,110,
%T 6,15,18,19,91,5,1,2,4,25,12,10,13,8,17,39,22,33,107,23,129,28,29,16,
%U 30,20,21,24,43,36,26,37,103,124,38,125,126,34,35,49,57,50,121
%N Irregular table of conjectural numbers. Let len(n) be the length (including n) of the Collatz (3x+1) iteration of n. The number n is in row n - len(n).
%C This sequence is labeled conjectural because (I think) we currently have no way of knowing whether there is some large n that has a long Collatz iteration. Sequence A220139 computes the trajectory for some large numbers, all of which have moderately small length (see A220140). The table below shows numbers in the same row. Notice that each row has only a few numbers. Row 15 is the first with 4 numbers: 26, 37, 103, 124. Row 324 is the first with 5 numbers: 336, 357, 370, 414, 440. Row 16028 has 6 numbers. Row 34937 has 7 numbers. No row in the first 10^7 rows has more than 7 numbers. This sequence is fairly linear; in the first 10^7 terms, no adjacent terms differ by more than 547.
%H T. D. Noe, <a href="/A228060/b228060.txt">Rows n = -85..1000 of irregular triangle, flattened</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CollatzProblem.html">Collatz Problem</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Collatz_conjecture">Collatz conjecture</a>
%e Example:
%e row -85: 27
%e row -76: 31
%e row -69: 41
%e row -59: 54
%e row -58: 47, 55
%e row -46: 62
%e row -45: 63
%e row -43: 73
%e row -32: 71
%e row -29: 82
%e row -28: 83
%e row -22: 97
%e row -12: 94
%e row -11: 9, 95
%e row -10: 7
%e row -6: 108
%e row -5: 3, 109
%e row -4: 11, 14, 110
%e row -3: 6, 15, 18
%e row -2: 19, 91
%e row -1: 5
%e row 0: 1, 2
%e row 1: 4, 25
%e row 2: 12
%e row 3: 10, 13
%e row 4: 8, 17, 39
%t nn = 100; Collatz[n_] := NestWhileList[If[EvenQ[#], #/2, 3 # + 1] &, n, # > 1 &]; t2 = Table[{n - Length[Collatz[n]], n}, {n, 2*nn}]; t3 = Select[t2, #[[1]] <= nn &]; Transpose[Sort[t3]][[2]]
%Y Cf. A070165 (Collatz trajectories).
%Y Cf. A008908 (length of trajectories).
%Y Cf. A082984, A228014.
%K nonn,tabf
%O -85,1
%A _T. D. Noe_, Aug 21 2013
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