A233293 - OEIS

%I #28 Jan 14 2014 10:09:38

%S 3,1,0,40,0,0,16,0,88,592,0,628,52,160,304,1672,808,2248,3616,11176,

%T 10096,8728,4192,23056,13912,65428,40804,5812,9448,12148,8584,82132,

%U 27700,10528,91672,53188,58804,20896,96064,2752,32776,25972,14560,183688,8080

%N Smallest number that is the largest value in the Collatz (3x + 1) trajectories of exactly n initial values. (a(n)=0 if no such number exists.)

%C Smallest number that appears exactly n times in A025586.

%C Numbers that are not the largest value in the 3x + 1 trajectory of any initial value (that is, numbers that do not appear at all in A025586) are in A213199; the smallest such number is a(0) = 3.

%C Numbers that are the largest value in the 3x + 1 trajectory of exactly one initial value (that is, numbers that appear exactly once in A025586) are in A222562; the smallest such number is a(1) = 1.

%C Numbers that are the largest value in the 3x + 1 trajectories of exactly three initial values (that is, numbers that appear exactly three times in A025586) are in A232870; the smallest such number is a(3) = 40.

%C No number that is the largest value in the 3x + 1 trajectories of exactly 2, 4, 5, 7, or 10 initial values exists, so a(n) = 0 at n = 2, 4, 5, 7, and 10; for all other values of n up to 3000, a(n) > 0. Conjecture: a(n) > 0 for all n > 10. - _Jon E. Schoenfield_, Dec 14 2013

%H Jon E. Schoenfield, <a href="/A233293/b233293.txt">Table of n, a(n) for n = 0..3000</a>

%e a(0) = 3 because no 3x + 1 trajectories have 3 as their largest value, and 3 is the smallest number for which this is the case.

%e a(1) = 1 because exactly one 3x + 1 trajectory (namely, the one whose initial value is 1) has 1 as its largest value (and 1 is the smallest number for which this is the case).

%e a(3) = 40 because exactly three 3x + 1 trajectories (the ones whose initial values are 13, 26, and 40) have 40 as their largest value, and 40 is the smallest number for which this is the case.

%e a(2) = 0 because there exists no number that is the largest value in exactly two 3x + 1 trajectories.

%t CollatzSeq[n_] := NestWhileList[If[EvenQ[#], #/2, 3 # + 1] &, n, # > 1 &]; nn = 250000; c = Table[Max[CollatzSeq[n]], {n, nn}]; srt = Tally[Sort[c]]; times = Transpose[srt][[2]]; u = Union[times]; d = Differences[u]; mx = Position[d, _?(# > 1 &), 1, 1][[1, 1]] - 1; u2 = Transpose[Take[srt, 10]][[1]]; t0 = Complement[Range[u2[[-1]]], u2][[1]]; t = Table[k = srt[[Position[times, n, 1, 1][[1, 1]]]]; If[k[[1]] > nn, 0, k[[1]]], {n, mx}]; t = Join[{t0}, t] (* _T. D. Noe_, Dec 07 2013 *)

%Y Cf. A025586, A213199, A222562, A232870.

%K nonn

%O 0,1

%A _Jon E. Schoenfield_, Dec 06 2013