cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A225124 a(n) = least number k such that 2^n is highest in Collatz(3x+1) trajectory of k.

Original entry on oeis.org

1, 2, 4, 8, 3, 32, 21, 128, 85, 512, 151, 2048, 1365, 8192, 5461, 32768, 14563, 131072, 87381, 524288, 349525, 2097152, 932067, 8388608, 5592405, 33554432, 22369621, 134217728, 26512143, 536870912, 357913941
Offset: 0

Views

Author

Jayanta Basu, Apr 29 2013

Keywords

Examples

			a(4)=3 since 3 is the least number such that largest member of Collatz(3 x + 1) trajectory of 3 is 2^4 = 16.

Crossrefs

Cf. A000079, A025586, A095381, A231610.

Programs

  • Haskell
    a225124 = (+ 1) . fromJust . (`elemIndex` a025586_list) . a000079
-- Reinhard Zumkeller, Apr 30 2013
  • Mathematica
    Coll[n_]:=NestWhileList[If[EvenQ[#],#/2,3*#+1]&,n,#>1 &]; t={}; Do[i=1; While[Max[Coll[i]] != 2^n, i++]; AppendTo[t, i], {n,0,25}]; t

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

Views

Author

Jon E. Schoenfield, Dec 01 2013

Keywords

Comments

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.

Examples

			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

Links

Crossrefs

Cf. A025586, A213199, A222562, A233293.

A341235 a(n) is the greatest term in n-th row of A341231.

Original entry on oeis.org

1, 2, 4, 4, 14, 6, 8, 8, 28, 14, 28, 12, 28, 14, 16, 16, 62, 28, 52, 20, 62, 28, 56, 24, 62, 28, 44, 28, 52, 30, 32, 32, 122, 62, 100, 36, 110, 52, 104, 40, 122, 62, 124, 44, 118, 56, 112, 48, 122, 62, 84, 52, 112, 54, 88, 56, 110, 58, 76, 60, 100, 62, 64, 64
Offset: 1

Views

Author

Rémy Sigrist, Feb 07 2021

Keywords

Comments

Records of a(n)/n appear to happen for n in A083318.

Examples

			For n = 10:
- the trajectory of 10 under A245471 is 10 -> 5 -> 14 -> 7 -> 8 -> 4 -> 2 -> 1,
- so a(10) = 14.

Links

Crossrefs

Cf. A025586, A083318, A245471, A340873, A341231.

Programs

  • PARI
    a(n) = { my (m=n); while (n>1, m=max(m, n=if (n%2, bitxor(n, 2*n+1), n/2))); m }

Formula

a(n) >= n, equality implies that n equals 1 or is even.
a(n) < 4*n.

A350806 The maximum value reached when starting at n for the iteration map A350805(n).

Original entry on oeis.org

1, 2, 1385, 4, 16, 160, 1385, 1385, 32, 1385, 160, 12, 268, 1385, 124, 16, 1385, 32, 268, 32, 32, 1385, 1385, 268, 32, 124, 30, 268, 124, 30, 1385, 32, 1385, 160, 1385, 1385, 124, 1385, 1385, 268, 124, 1385, 1385, 1385, 1385, 1385, 1385, 1385, 1385, 1385, 1385, 1385, 160, 1385, 1385, 56, 1385
Offset: 1

Views

Author

Scott R. Shannon, Jan 17 2022

Keywords

Comments

For smaller values of n a maximum value of 1385 is common; this is similar to the value 9232 for the standard Collatz map. See A025586.
A graph of the terms looks similar to those of A336957 and A098550. See the linked image.

Examples

			a(6) = 160 as the map A350805(6) gives 6 -> 11 -> 34 -> 53 -> 160 -> 21 -> 32 -> 5 -> 16 -> 4 -> 2 -> 1, the maximum value of which is 160.

Links

Crossrefs

Cf. A350805, A025586, A006370, A027746, A350807, A006577, A000040, A336957, A098550.

A063045 Largest value in Collatz-2 (A063041) trajectory starting at n.

Original entry on oeis.org

36, 36, 36, 36, 36, 36, 36, 36, 36, 144, 36, 144, 36, 36, 36, 5184, 36, 324, 36, 36, 144, 5184, 36, 36, 144, 36, 36, 121104, 36, 900, 36, 144, 5184, 36, 36, 5184, 324, 144, 40, 5184, 42, 1764, 144, 45, 5184
Offset: 2

Views

Author

Reinhard Zumkeller, Jul 07 2001

Keywords

Comments

a(47) currently unknown but large (see A280707). - Sean A. Irvine, Apr 16 2023

Examples

			a(3) = 36 (see A063042), a(17) = 5184 (see A063043), a(29) = 121104 (see A063044)

Crossrefs

Cf. A063041, A063042, A063043, A063044, A025586, A280707.

Extensions

a(40), a(42) and a(45) corrected by Sean A. Irvine, Apr 16 2023

A087225 Position of the largest peak value in Collatz (3x+1) trajectory.

Original entry on oeis.org

1, 1, 4, 1, 2, 5, 6, 1, 9, 3, 4, 6, 2, 7, 8, 1, 2, 10, 4, 1, 2, 5, 6, 1, 7, 3, 78, 8, 2, 9, 73, 1, 2, 3, 4, 11, 2, 5, 11, 1, 76, 3, 4, 6, 2, 7, 71, 1, 2, 8, 4, 1, 2, 79, 79, 1, 7, 3, 9, 10, 2, 74, 74, 1, 2, 3, 4, 1, 2, 5, 69, 1, 82, 3, 4, 6, 2, 12, 8, 1, 2, 77, 77
Offset: 1

Views

Author

Labos Elemer, Aug 27 2003

Keywords

Examples

			For n=9: iteration-list = [9, 28, 14, 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1]; peak = 52 at 9th position, so a(9) = 9.

Links

Crossrefs

Cf. A025586.

Programs

  • Mathematica
    Collatz[n_] :=NestWhileList[If[EvenQ[#], #/2, 3*# + 1] &, n, # > 1 &]; Flatten[Table[Position[Collatz[n], Max[Collatz[n]]], {n, 96}]] (* Jayanta Basu, Mar 24 2013 *)

A087253 Number of distinct initial values of various 3x+1 trajectories of which the peak-value equals 4n.

Original entry on oeis.org

1, 1, 0, 6, 1, 1, 0, 1, 0, 3, 0, 1, 12, 1, 0, 3, 1, 1, 0, 1, 1, 8, 0, 1, 3, 1, 0, 3, 1, 0, 0, 1, 1, 3, 0, 1, 3, 1, 0, 13, 0, 1, 0, 1, 1, 3, 0, 1, 8, 1, 0, 3, 1, 0, 0, 1, 1, 6, 0, 1, 3, 0, 0, 3, 1, 1, 0, 1, 1, 3, 0, 1, 0, 1, 0, 14, 1, 1, 0, 1, 1, 0, 0, 1, 6, 1, 0, 3, 1, 1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 3, 1, 1, 0, 1, 0
Offset: 1

Views

Author

Labos Elemer, Sep 08 2003

Keywords

Examples

			n=13: a(13)=12 because peak=4.13=52
occurs at the following 12 initial values:
{7,9,11,14,17,18,22,28,34,36,44,52};
this list differs from trajectory started at iv=52.
Impossible peaks of 4k form are listed in A087252
(corresponding to a[x]=0).

Crossrefs

Cf. A025586.

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

Views

Author

Labos Elemer, Sep 08 2003

Keywords

Comments

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

Examples

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

Crossrefs

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

Programs

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

A087255 Number of different initial values for 3x+1 trajectories of which the largest term appearing during the iteration equals n.

Original entry on oeis.org

1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 12, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 3, 0, 0, 0, 1, 0
Offset: 1

Views

Author

Labos Elemer, Sep 08 2003

Keywords

Examples

			a(2k+1)=a(4k+2)=a(16k+12)=0 because larger values appear in iteration: 6k+4, 6k+4, and 18k+16, respectively. Exception: a(1)=a(2)=1.
Some outstanding cases: a(9232)=1579, a(13120)=75.

Links

Crossrefs

Cf. A025586, A087251-A087254.

Extensions

Edited by Jon E. Schoenfield, Dec 01 2013

A095382 Exponents of power of 2 of the largest terms arising in the 3x+1 iterations started with terms of A095381.

Original entry on oeis.org

0, 1, 4, 2, 4, 4, 3, 4, 4, 4, 6, 5, 6, 6, 8, 7, 10, 8, 10, 10, 8, 10, 10, 10, 10, 9, 10, 10, 10, 10, 10, 12, 11, 12, 12, 14, 13, 14, 16, 14, 16, 16, 16, 15, 16, 16, 16, 16, 18, 17, 18, 18, 20, 19, 20, 22, 20, 22, 22, 21, 22, 22, 22
Offset: 1

Views

Author

Labos Elemer, Jun 14 2004

Keywords

Crossrefs

Cf. A025586, A087256, A095381.

Formula

a[n]=Log[2, A025586[A095381(n)]]
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