A233293 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.)

3, 1, 0, 40, 0, 0, 16, 0, 88, 592, 0, 628, 52, 160, 304, 1672, 808, 2248, 3616, 11176, 10096, 8728, 4192, 23056, 13912, 65428, 40804, 5812, 9448, 12148, 8584, 82132, 27700, 10528, 91672, 53188, 58804, 20896, 96064, 2752, 32776, 25972, 14560, 183688, 8080

Offset

0,1

Comments

Smallest number that appears exactly n times in A025586.

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.

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.

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.

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

Links

Jon E. Schoenfield, Table of n, a(n) for n = 0..3000

Example

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

Mathematica

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 *)

Crossrefs

Cf. A025586, A213199, A222562, A232870.

Sequence in context: A158782 A187558 A327547 * A361579 A066746 A278105

Adjacent sequences: A233290 A233291 A233292 * A233294 A233295 A233296

Keyword

nonn

Author

Jon E. Schoenfield, Dec 06 2013

Status

approved