A025586 -id:A025586 - cp's OEIS frontend

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

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 *)

A338196 The integers sorted according to the maximum number in their Collatz (3*x+1) orbit, and in the case of a tie, sorted according to their value.

Original entry on oeis.org

1, 2, 4, 8, 3, 5, 6, 10, 12, 16, 20, 24, 32, 13, 26, 40, 48, 7, 9, 11, 14, 17, 18, 22, 28, 34, 36, 44, 52, 56, 21, 42, 64, 68, 72, 80, 84, 19, 25, 29, 38, 50, 58, 76, 88, 96, 33, 66, 100, 104, 37, 74, 112, 116, 128, 132, 45, 90, 136, 144, 49, 98, 148, 152, 15

Offset: 1

Hugo Pfoertner, Oct 21 2020

nonn

Assuming the validity of the Collatz conjecture, this sequence is a permutation of the natural numbers.

			a(1) = 1, a(2) = 2, a(3) = 4, a(4) = 8 are already the maximum in their orbit.
a(5) = 3, a(6) = 5, a(7) = 6, a(8) = 10, a(9) = 12, and a(10) = 16 all have an orbit with 16 as maximum and are therefore sorted by their size.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program for A338196

Cf. A006370, A006577, A025586, A087255.

PARI
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See Links section.
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A375955 T(n,k) for n >= 1, k <= n is the maximum value in the intersection of the Collatz trajectories of n and k, where a trajectory ends when it reaches 1. T(n,k) is a triangle read by rows.

Original entry on oeis.org

1, 1, 2, 1, 2, 16, 1, 2, 4, 4, 1, 2, 16, 4, 16, 1, 2, 16, 4, 16, 16, 1, 2, 16, 4, 16, 16, 52, 1, 2, 8, 4, 8, 8, 8, 8, 1, 2, 16, 4, 16, 16, 52, 8, 52, 1, 2, 16, 4, 16, 16, 16, 8, 16, 16, 1, 2, 16, 4, 16, 16, 52, 8, 52, 16, 52, 1, 2, 16, 4, 16, 16, 16, 8, 16, 16, 16, 16

Offset: 1

Markus Sigg, Sep 03 2024

			The triangle begins:
       k=1  2   3  4   5   6    7  8
  n=1:   1;
  n=2:   1, 2;
  n=3:   1, 2, 16;
  n=4:   1, 2,  4, 4;
  n=5:   1, 2, 16, 4, 16;
  n=6:   1, 2, 16, 4, 16, 16;
  n=7:   1, 2, 16, 4, 16, 16, 52;
  n=8:   1, 2,  8, 4,  8,  8,  8, 8;
  ...
T(20,3) = 16 since the trajectory of 20 is (20,10,5,16,8,4,2,1), the trajectory of 3 is (3,10,5,16,8,4,2,1), and their intersection has the maximum 16. This example shows that T(n,k) does not necessarily denote the start of the common trajectory of n and k.

Markus Sigg, Table of n, a(n) for n = 1..11325 (row 1..150).

Cf. A025586 (main diagonal)

C(n) = my(L = List([n])); while(n > 1, n = if(n % 2 == 0, n/2, 3*n + 1); listput(L, n)); Set(L);
a375955_row(n) = my(Cn = C(n)); vector(n, k, vecmax(setintersect(Cn, C(k))));

T(n,n) = A025586(n).

A376079 a(n) is the largest difference of adjacent elements in the sorted list of the Collatz trajectory elements of n.

Original entry on oeis.org

1, 6, 2, 8, 6, 12, 4, 12, 6, 12, 4, 20, 12, 54, 8, 14, 12, 30, 6, 32, 12, 54, 8, 18, 14, 1944, 12, 36, 54, 1944, 16, 18, 12, 54, 12, 56, 30, 66, 20, 1944, 22, 48, 8, 68, 54, 1944, 24, 38, 18, 78, 14, 80, 1944, 1944, 12, 24, 30, 66, 54, 54, 1944, 1944, 32, 48, 12

Offset: 2

Markus Sigg, Sep 09 2024

nonn

Trajectories end when they reach 1.

			The trajectory of 3 is (3,10,5,16,8,4,2,1), the sorted list of the trajectory elements is (1,2,3,4,5,8,10,16), the list of differences is (1,1,1,1,3,2,6) with maximum 6, so a(3) = 6.

Markus Sigg, Table of n, a(n) for n = 2..10001

Cf. A220237 (sorted trajectory), A008908, A025586.

Table[Max[Differences[Sort[NestWhileList[If[EvenQ[#],#/2,3#+1]&,n,#>1&]]]],{n,2,70}] (* Harvey P. Dale, Aug 24 2025 *)

a(n) = my(L = List([n])); while(n > 1, n = if(n % 2 == 0, n/2, 3*n + 1); listput(L, n)); listsort(L); vecmax(vector(#L - 1, i, L[i+1] - L[i]));

A087264 a(n) is the least initial value of a 3x+1 trajectory in which 4n is the largest(peak) term or a(n)=0 if 4n cannot be a peak.

Original entry on oeis.org

4, 8, 0, 3, 20, 24, 0, 32, 0, 13, 0, 48, 7, 56, 0, 21, 68, 72, 0, 80, 84, 19, 0, 96, 33, 104, 0, 37, 116, 0, 0, 128, 132, 45, 0, 144, 49, 152, 0, 15, 0, 168, 0, 176, 180, 61, 0, 192, 43, 200, 0, 69, 212, 0, 0, 224, 228, 51, 0, 240, 81, 0, 0, 85, 260, 264, 0, 272, 276, 93, 0, 288

Offset: 1

Labos Elemer, Sep 11 2003

nonn

			n=10: 4n=40 arises as a peak-term in 3x+1 trajectory started at 13, so a[10]=13 and 13 is the least initial value including 40 as a max-term.

Cf. A025586, A087263.

c[x_] := (1-Mod[x, 2])*(x/2)+Mod[x, 2]*(3*x+1); c[1]=1; fpl[x_] := Max[Delete[FixedPointList[c, x], -1]] t=Table[fpl[w], {w, 1, 15000}]; Table[Min[Flatten[Position[t, 4*j]]], {j, 1, 256}]

A095385 Numbers k with property that k is a peak value in 3x+1 trajectory such that both k+1 and k-1 are prime numbers.

Original entry on oeis.org

4, 72, 180, 192, 228, 240, 312, 600, 1092, 1152, 1428, 1488, 1608, 1620, 1872, 2088, 2112, 2592, 2688, 3000, 3168, 3252, 3360, 3528, 4272, 4548, 4800, 5640, 6552, 6960, 7488, 7560, 8088, 8292, 8388, 8820, 9000, 9012, 9240, 9768, 10008, 10068, 10272

Offset: 1

Labos Elemer, Jun 14 2004

nonn

In such cases it seems that initial value equals peak value. Proof needed!

			k=72: list={72,36,18,9,28,14,7,22,11,34,17,52,26,13,40,20,10,5,16,8,4,2,1}

Intersection of A014574 and A025586.

c[x_]:=c[x]=(1-Mod[x, 2])*(x/2)+Mod[x, 2]*(3*x+1);c[1]=1; fpl[x_]:=FixedPointList[c, x]; {$RecursionLimit=1000;m=0}; Do[If[PrimeQ[1+Max[fpl[n]]]&&PrimeQ[ -1+Max[fpl[n]]], Print[n]], {n, 1, 10000}]

A095386 Largest prime factor of peak values of 3x+1 trajectory started at n.

Original entry on oeis.org

2, 2, 2, 2, 2, 13, 2, 13, 2, 13, 2, 5, 13, 5, 2, 13, 13, 11, 5, 2, 13, 5, 3, 11, 5, 577, 13, 11, 5, 577, 2, 5, 13, 5, 13, 7, 11, 19, 5, 577, 2, 7, 13, 17, 5, 577, 3, 37, 11, 29, 13, 5, 577, 577, 7, 7, 11, 19, 5, 23, 577, 577, 2, 7, 5, 19, 17, 13, 5, 577, 3, 577, 7, 17, 11, 29, 19, 101, 5

Offset: 2

Labos Elemer, Jun 14 2004

nonn

			n=27: peak = 9232 = 2*2*2*2*577, so a(27)=577;
more extensive search suggests that all primes may occur as largest prime factor of peak.

Cf. A006530, A025586.

a(n) = A006530(A025586(n)).

A095388 Smallest multiple of 2^n whose Collatz (3x+1) trajectory includes at least one larger number.

Original entry on oeis.org

6, 12, 120, 432, 864, 1728, 3456, 6912, 931328, 4357120, 19789824, 249753600, 499507200, 1272561664, 5226070016, 10452140032, 351882051584, 1215818366976, 3364158439424, 6953815244800, 13907630489600, 27815260979200, 55630521958400, 1343005923475456

Offset: 1

Labos Elemer, Jun 14 2004

nonn

Although the Collatz trajectory of a multiple of 2^n begins with n consecutive halving steps, some such trajectories nevertheless reach a larger peak value.

			The Collatz trajectory of 120 = 2^3 * 15 begins with {120, 60, 30, 15, 46, 23, 70, 35, 106, 53, 160, ...}, and 160 > 120, and there is no number k < 120 of the form 2^3 * m whose trajectory includes a number > k, so a(3) = 120.

Cf. A025586.

c[x_]:=c[x]=(1-Mod[x, 2])*(x/2)+Mod[x, 2]*(3*x+1);c[1]=1; fpl[x_]:=Delete[FixedPointList[c, x], -1] {k=65536, ta=Table[0, {100}], u=1}; {$RecursionLimit=1000;m=0}; Do[If[Greater[Max[fpl[k*n]], k*n], Print[{k*n, n}]; ta[[u]]=k*n;u=u+1], {n, 1, 1000000}] [Code for 2^16 divisor, a(16)].

a(17)-a(24) from Donovan Johnson, Feb 02 2011

Edited by Jon E. Schoenfield, May 18 2024

A225038 Numbers n such that at least one member of Collatz (3x+1) trajectory of n is >= n^2.

Original entry on oeis.org

1, 3, 7, 27, 31, 41, 47, 54, 55, 62, 63, 71, 73, 82, 83, 91, 94, 95, 6631675, 7460635, 319804831, 379027947, 426406441, 479707247, 568541921, 598957743, 639609662, 639609663, 719560871, 758055894, 758055895, 852812882, 852812883, 898436615, 959414494, 959414495, 1010741193, 1079341307, 1137083842, 1137083843, 1410123943

Offset: 1

Jayanta Basu, Apr 25 2013

nonn

Many of these numbers are on the same trajectory. For instance, the numbers from 27 to 95 are all on the Collatz trajectories of 27 and 54. See Roosendaal's web page for more possibilities. - T. D. Noe, Apr 25 2013

			3 is a member since both 16 and 10 both belong to Collatz trajectory of 3 that are >= 3^2 = 9.

Eric Roosendaal, 3x+1 path records

Cf. A025586, A070165.

Coll[n_] := NestWhileList[If[EvenQ[#], #/2, 3 # + 1] &, n, # > 1 &]; t = {}; Do[If[Max[Coll[n]] >= n*n, AppendTo[t, n]], {n, 1000}]; t

Numbers n such that A025586(n) >= n^2.

Extended by T. D. Noe, Apr 25 2013

A341362 a(n) begins the first sequence of n consecutive positive integers with the same h-value and the same d-value in the Collatz (or '3x + 1') problem.

Original entry on oeis.org

1, 54, 108, 290, 290, 386, 172146, 298200, 596400, 596400, 596400, 795201, 795201, 2849196, 2849196, 8965036, 33659819, 45529226, 52417676, 93186987, 104161282, 104161282, 104161282, 436089218, 436089218, 605581697, 934358530, 934358530, 934358530, 3826876112

Offset: 1

Lamine Ngom, Feb 10 2021

nonn

The Collatz function is as follows: F(x) = x/2 if x is even, otherwise F(x) = 3*x+1.
It is conjectured that starting from any number, and repeatedly applying the function on its previous result, we will always reach 1.
The d-value (or flight duration, A006577) is the number of steps needed to reach 1.  The h-value (or flight height, A025586) is the maximum of the number's trajectory.

			a(3) = 108 because 108, 109 and 110 have same d-value (113) and same h-value (9232).
And 108 is the smallest number starting such sequence of 3 consecutive positive integers with same d-value and same h-value.

Guo-Gang Gao, On consecutive numbers of the same height in the Collatz problem, Discrete Mathematics, Volume 112, pages 261-267, 1993.

Cf. A078441, A277109, A268253, A006577 (duration), A025586 (height), A056959.

cp's OEIS Frontend

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

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A338196 The integers sorted according to the maximum number in their Collatz (3*x+1) orbit, and in the case of a tie, sorted according to their value.

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A375955 T(n,k) for n >= 1, k <= n is the maximum value in the intersection of the Collatz trajectories of n and k, where a trajectory ends when it reaches 1. T(n,k) is a triangle read by rows.

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A376079 a(n) is the largest difference of adjacent elements in the sorted list of the Collatz trajectory elements of n.

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A087264 a(n) is the least initial value of a 3x+1 trajectory in which 4n is the largest(peak) term or a(n)=0 if 4n cannot be a peak.

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A095385 Numbers k with property that k is a peak value in 3x+1 trajectory such that both k+1 and k-1 are prime numbers.

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A095386 Largest prime factor of peak values of 3x+1 trajectory started at n.

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A095388 Smallest multiple of 2^n whose Collatz (3x+1) trajectory includes at least one larger number.

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Extensions

A225038 Numbers n such that at least one member of Collatz (3x+1) trajectory of n is >= n^2.

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