A222562 -id:A222562 - 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 *)

A213199 Numbers k whose Collatz (3x+1) trajectory includes at least one term that is greater than k.

Original entry on oeis.org

3, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 21, 22, 23, 25, 26, 27, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 53, 54, 55, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 81, 82, 83

Offset: 1

Jayanta Basu, Mar 01 2013

nonn

			a(1) = 3 because 3 is the first number k whose trajectory goes above k.

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A033496, A222562, A222597.

Collatz[n_] := NestWhileList[If[EvenQ[#], #/2, 3 # + 1] &, n, # > 1 &]; Select[Range[100], Max[Collatz[#]] > # &] (* T. D. Noe, Mar 01 2013 *)

A232870 Numbers that are the largest value in the Collatz (3x+1) trajectories of exactly three initial values.

Original entry on oeis.org

40, 64, 100, 112, 136, 148, 184, 208, 244, 256, 280, 352, 400, 424, 472, 532, 544, 616, 640, 688, 712, 724, 784, 820, 832, 868, 904, 928, 964, 976, 1048, 1072, 1108, 1120, 1156, 1192, 1216, 1264, 1300, 1360, 1396, 1408, 1432, 1480, 1540, 1576, 1588, 1624, 1684

Offset: 1

Jon E. Schoenfield, Dec 01 2013

nonn

Numbers that appear exactly 3 times in A025586, which gives the largest value in the 3x + 1 trajectory of n.
For each term k in this sequence, the three initial values, that is, values of n at which A025586(n) = k, are (in ascending order) n1 = (k-1)/3, n2 = 2*n1 = 2*(k-1)/3, and n3 = k. n1 is the odd number from which an upward (that is, 3x + 1) step lands at k = 3*n1 + 1. It cannot be the case that n1 = 3 (mod 4), because we would then have k = 10 (mod 12), so k/2 would be odd, and its successor in the trajectory would be 3*k/2 + 1 > k, so k would not be the largest value in the trajectory. Thus, n1 = 1 (mod 4), so n2 = 2 (mod 8) and n3 = 4 (mod 12).
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.
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.

			40 is in the sequence because it is the largest value in the 3x + 1 trajectories of exactly three initial values: 13, 26, and 40 itself. The trajectories are as follows:
..... 13 -> 40 -> 20 -> 10 -> 5 -> 16 -> 8 -> 4 -> 2 -> 1
26 -> 13 -> 40 -> 20 -> 10 -> 5 -> 16 -> 8 -> 4 -> 2 -> 1
........... 40 -> 20 -> 10 -> 5 -> 16 -> 8 -> 4 -> 2 -> 1

Jon E. Schoenfield, Table of n, a(n) for n = 1..10000

Cf. A025586, A213199, A222562, A233293.

A087254 If we start the Collatz-iteration at these values, each divisible by 4, all subsequent terms in trajectory are smaller than the initial value.

Original entry on oeis.org

4, 8, 20, 24, 32, 48, 56, 68, 72, 80, 84, 96, 104, 116, 128, 132, 144, 152, 168, 176, 180, 192, 200, 212, 224, 228, 240, 260, 264, 272, 276, 288, 296, 308, 312, 320, 324, 336, 344, 356, 360, 368, 372, 384, 392, 404, 408, 416, 452, 456, 464, 468, 480, 488, 512

Offset: 1

Labos Elemer, Sep 08 2003

nonn

Numbers that are not highest in any Collatz trajectory other than n. - Jayanta Basu, May 27 2013

			n=104: iteration list = {104,52,26,13,40,20,10,5,16,8,4,2,1}, where initial-value = largest-term.

Cf. A025586, A087251-A087253, A222562 (with 1 and 2 prepended).

mcoll[n_]:=Max@@NestWhileList[If[EvenQ[#],#/2,3#+1] &,n,#>1 &]; t={}; Do[c=i=0; While[c!=1 && ++iJayanta Basu, May 27 2013 *)

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.)

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A213199 Numbers k whose Collatz (3x+1) trajectory includes at least one term that is greater than k.

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A232870 Numbers that are the largest value in the Collatz (3x+1) trajectories of exactly three initial values.

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A087254 If we start the Collatz-iteration at these values, each divisible by 4, all subsequent terms in trajectory are smaller than the initial value.

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