A232870 -id:A232870 - cp's OEIS frontend

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

Jon E. Schoenfield, Dec 06 2013

nonn

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

			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.

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

Cf. A025586, A213199, A222562, A232870.

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 *)

A274467 Numbers that are the largest value in the Collatz (3x+1) trajectories of exactly six initial values.

Original entry on oeis.org

16, 232, 340, 448, 1204, 1636, 1960, 2176, 2500, 2608, 3256, 3472, 3688, 3796, 3904, 4336, 4552, 4768, 5092, 5200, 5416, 5632, 5956, 6064, 6496, 6928, 7252, 7360, 7576, 8116, 8548, 8656, 8872, 8980, 9304, 9412, 9520, 9736, 9952, 10168, 10384, 10600, 10708, 10816, 11032, 11464, 11572, 11680

Offset: 1

Hartmut F. W. Hoft, Jun 24 2016

nonn

Numbers that appear exactly 6 times in A025586, which gives the largest value in the 3x + 1 trajectory of n. This sequence is a subsequence of A033496 and also of A176869.

There is a single Collatz trajectory containing all initial values to its maximum value n which has the form (8n-20)/9, (4n-10)/9, (2n-5)/9, (2n-2)/3, (n-1)/3, n, where n mod 3 = 1, (2n-2)/3 mod 3 = 1, (4n-10)/9 mod 3 = 0; see also the link in A033496.

			1636 is in the sequence since it is the largest value in the single trajectory starting with 1452, 726, 363, 1090, 545, 1636, and no other initial values produce a trajectory with maximum 1636.

Hartmut F. W. Hoft, Table of n, a(n) for n = 1..5320

Cf. A025586, A033496, A176869, A232870.

(* function fanSize[] is defined in A105730 *)
a274467[low_, high_] := First[Transpose[Select[Map[{#, fanSize[#]}&, Range[low, high, 4]], Last[#]==6&]]]/; Mod[low, 4]==0
a274467[4,10000] (* Data *)

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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.)

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