A127824 -id:A127824 - cp's OEIS frontend

A341220 Irregular table read by rows T(n, k), n >= 0, k = 1..A341218(n); T(n, k) is the k-th number m such that A340873(m) = n.

1, 2, 4, 3, 8, 6, 7, 16, 12, 14, 15, 32, 5, 24, 28, 30, 31, 64, 10, 11, 48, 56, 60, 62, 63, 128, 20, 21, 22, 23, 96, 112, 120, 124, 126, 127, 256, 13, 40, 42, 43, 44, 46, 47, 192, 224, 240, 248, 252, 254, 255, 512, 25, 26, 27, 80, 84, 85, 86, 87, 88, 92, 94, 95, 384, 448, 480, 496, 504, 508, 510, 511, 1024

Offset: 0

Rémy Sigrist, Feb 07 2021

Each positive integer appears once.
For any n >= 0:
- if m appears in row n, then 2*m appears in row n+1,
- if m appears in row n and is an odious even number > 2, then an additional odd number appears in row n+1.

			Table begins:
    1;
    2;
    4;
    3, 8;
    6, 7, 16;
    12, 14, 15, 32;
    5, 24, 28, 30, 31, 64;
    10, 11, 48, 56, 60, 62, 63, 128;
    20, 21, 22, 23, 96, 112, 120, 124, 126, 127, 256;
    13, 40, 42, 43, 44, 46, 47, 192, 224, 240, 248, 252, 254, 255, 512;
    ...

Rémy Sigrist, Table of n, a(n) for n = 0..11792 (rows 0..26)
Rémy Sigrist, PARI program for A341220
Index entries for sequences related to 3x+1 (or Collatz) problem
Index entries for sequences that are permutations of the natural numbers

Cf. A127824, A340873, A341194, A341218.

PARI
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See Links section.
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T(n, 1) = A341194(n).

T(n, A341218(n)) = 2^n.

A375094 a(n) is the least number not occurring in a Collatz trajectory of n steps.

Original entry on oeis.org

2, 3, 3, 3, 3, 3, 3, 6, 7, 7, 7, 7, 7, 7, 7, 7, 9, 9, 9, 18, 25, 25, 25, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27

Offset: 0

Markus Sigg and Hugo Pfoertner, Aug 03 2024

A006877 and A288493 form a run-length encoding of this sequence: It starts with A288493(1) copies of A006877(2), followed by A288493(2) copies of A006877(3), followed by A288493(3) copies of A006877(4), and so on.

			a(5) = 3 because there are two trajectories with 5 steps, namely (32,16,8,4,2,1) and (5,16,8,4,2,1). 3 is the smallest number not appearing in both.

Markus Sigg and Hugo Pfoertner, Table of n, a(n) for n = 0..2441
Wikipedia, Collatz Conjecture
Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A005186, A006877, A127824, A288493.

# output in b-file format
from itertools import count
n = 0
for k in count():
    m = k
    s = 0
    while m > 1:
        m = m // 2 if m % 2 == 0 else 3*m+1
        s += 1
    while n < s:
        print(n, k, flush=True)
        n += 1

A088976 Breadth-first traversal of the Collatz tree, with the even child of each node traversed prior to its odd child. If the Collatz 3n+1 conjecture is true, this is a permutation of all positive integers.

Original entry on oeis.org

1, 2, 4, 8, 16, 32, 5, 64, 10, 128, 21, 20, 3, 256, 42, 40, 6, 512, 85, 84, 80, 13, 12, 1024, 170, 168, 160, 26, 24, 2048, 341, 340, 336, 320, 53, 52, 48, 4096, 682, 680, 113, 672, 640, 106, 104, 17, 96, 8192, 1365, 1364, 227, 1360, 226, 1344, 1280, 213, 212, 35, 208

Offset: 0

David Eppstein, Oct 31 2003

T. D. Noe, Table of n, a(n) for n = 0..3517

Cf. A127824 (terms at the same depth are sorted).

def A088976():
    yield 1
    for x in A088976():
        yield 2*x
        if x > 4 and x % 6 == 4:
            yield (x - 1)//3
a = A088976(); print([next(a) for _ in range(100)])

A331272 Irregular triangle in which row n lists numbers m such that A330073(m,n) = 1.

Original entry on oeis.org

1, 5, 25, 4, 125, 20, 625, 3, 100, 104, 3125, 15, 16, 500, 520, 15625, 2, 75, 80, 83, 86, 2500, 2600, 2604, 78125, 10, 12, 13, 375, 400, 415, 416, 430, 433, 12500, 13000, 13020, 390625, 50, 60, 62, 65, 66, 69, 71, 1875, 2000, 2075, 2080, 2083, 2150, 2165, 2166, 62500, 65000, 65100, 65104, 1953125

Offset: 0

Davis Smith, Jan 13 2020

The last number in row n is 5^n (A000351(n)).
For numbers m and n such that m is in row n, 5*m is in row n + 1 and if m > 10 and m == 10, 15, 20, or 25 (mod 30), then floor(m/6) is in row n + 1.
The conjecture in A330073 claims that every positive integer appears in this triangle.

			The irregular triangle starts:
0:   1
1:   5
2:  25
3:   4  125
4:  20  625
5:   3  100  104 3125
6:  15   16  500  520 15625
7:   2   75   80   83    86  2500  2600  2604 78125
8:  10   12   13  375   400   415   416   430   433 12500 13000 13020 390625

T. Ahmed and H. Snevily, Are There an Infinite Number of Collatz Integers?, 2013.
M. Bruschi, A generalization of the Collatz problem and conjecture, arXiv:0810.5169 [math.NT], 2008.
W. Carnielli, Some Natural Generalizations Of The Collatz Problem, Applied Mathematics E-Notes, 15 (2015), 197-206.

Cf. A000351 (5^n), A127824, A330073.

A331272(lim)=my(N=[1], b=-1, RC=5*[2..5]); while(bif(setsearch(RC,X%30)&&(X>RC[1]),[floor(X/6),5*X],X*5),N))[1,]))

If N is the list of numbers in row n, then the list of numbers in row n + 1 is the union of each number in N multiplied by 5 and numbers floor(x/6) where x is in N, congruent to 0 (mod 5), not congruent to 0 or 5 (mod 30), and floor(x/6) > 1.

A337673 a(n) is the sum of all positive integers whose Collatz orbit has length n.

Original entry on oeis.org

0, 1, 2, 4, 8, 16, 37, 74, 172, 344, 786, 1572, 3538, 7206, 16252, 33112, 73762, 149967, 330107, 678610, 1498356, 3082302, 6742487, 13855154, 30122440, 62388962, 135783788, 281177482, 608402189, 1259151448, 2711432766, 5646008216, 12172417990, 25339969480, 54409676729, 113159496364

Offset: 0

Markus Sigg, Sep 15 2020

nonn

a(n) >= 2^(n-1) as 2^(n-1) has orbit length n.

			a(6) = 5+32 = 37 as the positive integers whose Collatz orbit has length 6 are {5,32} - the orbit of 5 is 5,16,8,4,2,1, and the orbit of 32 is 32,16,8,4,2,1.

Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A000975, A005186, A153772.

Equals row sums of triangles A088975 and A127824.

nextSet(s) = { my(s1 = Set([])); for(i = 1, #s, s1 = setunion(s1, Set([2*s[i]])); if (s[i] > 4 && (s[i]-1) % 3 == 0 && (s[i]-1)/3 % 2 == 1, s1 = setunion(s1, Set([(s[i]-1)/3]))); ); return(s1); }
a(n) = { my(s = Set([1])); for(k = 1, n, s = nextSet(s); ); return(sum(i=1,#s,s[i])); }

More terms from David A. Corneth, Sep 15 2020

A355312 Irregular triangle read by rows, in which the rows list groups of consecutive integers taking the same number of halving and tripling steps to reach 1 in '3X+1' problem. Groups are in order of the number of steps required, and in numerical order among those with the same number of steps.

Original entry on oeis.org

20, 21, 12, 13, 84, 85, 52, 53, 340, 341, 34, 35, 212, 213, 226, 227, 1364, 1365, 68, 69, 70, 452, 453, 454, 22, 23, 140, 141, 150, 151, 852, 853, 908, 909, 5460, 5461, 44, 45, 46, 276, 277, 300, 301, 302, 1812, 1813, 14, 15, 92, 93, 564, 565, 604, 605, 3412, 3413

Offset: 1

Paul Duckett, Jun 27 2022

			20 and 21 are terms since they both use 7 steps (trajectories 20, 10, 5, 16, 8, 4, 2, 1 and 21, 64, 32, 16, 8, 4, 2, 1).
300, 301 and 302 are terms since they all use 16 steps.
The triangle starts:
   7:  20   21
   9:  12   13   84   85
  11:  52   53  340  341
  13:  34   35  212  213  226  227 1364 1365
  14:  68   69   70  452  453  454
  15:  22   23  140  141  150  151  852  853  908  909 5460 5461
  16:  44   45   46  276  277  300  301  302 1812 1813
  17:  14   15   92   93  564  565  604  605 3412 3413

Rémy Sigrist, Table of n, a(n) for n = 1..11035
Rémy Sigrist, PARI program
Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A006577, A078417.

Subtriangle of A127824.

PARI
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See Links section.
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A177001 The number of 3x+1 steps in the Collatz iteration of A033491(n), the least number requiring n iterations.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 2, 2, 2, 2, 2, 3, 3, 4, 4, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 8, 8, 8, 9, 9, 9, 10, 10, 11, 11, 11, 11, 12, 12, 12, 13, 13, 13, 14, 14, 15, 15, 15, 16, 16, 16, 17, 17, 18, 18, 18, 18, 19, 19, 20, 20, 20, 21, 21, 22, 22, 23, 23, 24, 24, 24, 24, 25, 25, 25, 26, 26, 27, 27, 28

Offset: 1

T. D. Noe, Apr 30 2010

nonn

It appears that a(n) is the maximum number of 3x+1 steps for all numbers requiring n Collatz iterations, which is row n of A127824.

			24 is the smallest number that requires 10 Collatz iterations. The iteration uses two 3x+1 steps to produce 24, 12, 6, 3, 10, 5, 16, 8, 4, 2, 1. Hence a(10)=2.

col[n_] := NestWhileList[If[EvenQ[#], #/2, 3 # + 1] &, n, # > 1 &]; Table[k = 1; While[Length[(y = col[k])] - 1 != n, k++]; Count[y, ?OddQ] - 1, {n, 80}] (* _Jayanta Basu, Jul 27 2013 *)

For large n, a(n) ~ n * log(2)/log(6).

cp's OEIS Frontend

A341220 Irregular table read by rows T(n, k), n >= 0, k = 1..A341218(n); T(n, k) is the k-th number m such that A340873(m) = n.

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A375094 a(n) is the least number not occurring in a Collatz trajectory of n steps.

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A088976 Breadth-first traversal of the Collatz tree, with the even child of each node traversed prior to its odd child. If the Collatz 3n+1 conjecture is true, this is a permutation of all positive integers.

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A331272 Irregular triangle in which row n lists numbers m such that A330073(m,n) = 1.

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A337673 a(n) is the sum of all positive integers whose Collatz orbit has length n.

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A355312 Irregular triangle read by rows, in which the rows list groups of consecutive integers taking the same number of halving and tripling steps to reach 1 in '3X+1' problem. Groups are in order of the number of steps required, and in numerical order among those with the same number of steps.

Views

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Examples

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Crossrefs

Programs

PARI

A177001 The number of 3x+1 steps in the Collatz iteration of A033491(n), the least number requiring n iterations.

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