A176869 -id:A176869 - cp's OEIS frontend

A222562 Numbers that are highest in their respective Collatz (3x+1) trajectories only.

1, 2, 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

Offset: 1

Jayanta Basu, Feb 27 2013

nonn

This is effectively the complement of A176869 in A033496, excluding numbers which are also highest in trajectories less than the number itself.

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

Cf. A033496, A176869.

Collatz[n_] := NestWhileList[If[EvenQ[#], #/2, 3 # + 1] &, n, # > 1 &]; oldMax = {}; t = {}; Do[c = Collatz[n]; If[! MemberQ[oldMax, n] && Max[c] == n, AppendTo[t, n]]; oldMax = Union[oldMax, {Max[c]}], {n, 416}]; t (* T. D. Noe, Feb 28 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 *)

A375937 Odd numbers which are the largest odd number in their Collatz trajectory.

Original entry on oeis.org

1, 5, 13, 17, 21, 29, 33, 37, 45, 49, 53, 61, 65, 69, 77, 81, 85, 93, 101, 113, 117, 133, 141, 149, 157, 173, 177, 181, 197, 205, 209, 213, 229, 237, 241, 245, 261, 269, 273, 277, 289, 301, 305, 309, 317, 321, 325, 341, 349, 357, 369, 373, 385, 397, 401, 405

Offset: 1

Markus Sigg, Sep 03 2024

nonn

a(n) == 1 (mod 4) because the trajectory of 4x+3 is (4x+3, 12x+10, 6x+5, ...) and 6x+5 > 4x+3.

			The odd elements of the Collatz trajectory (3,10,5,16,8,4,2,1) are {3,5,1} with maximum 5 > 3, so 3 is not a term. The odd elements of the Collatz trajectory (13,40,20,10,5,16,8,4,2,1) are {13,5,1} with maximum 13, so 13 is a term.

Markus Sigg, Table of n, a(n) for n = 1..10000

Cf. A033496, A176869.

makeEntries(count) = {
    my(L = List(), k = 1);
    while(#L < count,
        my(m = k);
        while(m > 1 && m <= k,
            m = 3*m + 1;
            while(m % 2 == 0, m = m / 2);
        );
        if(m == 1, listput(L, k));
        k += 2
    );
    L
};
print(Vec(makeEntries(56)));

a(n) = (A176869(n) - 1) / 3 for n > 1.

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A222562 Numbers that are highest in their respective Collatz (3x+1) trajectories only.

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

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A375937 Odd numbers which are the largest odd number in their Collatz trajectory.

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