A070167 -id:A070167 - cp's OEIS frontend

A359226 a(n) is the least k >= 0 such that A006370^k(A070167(n)) = n (where A006370^k denotes the k-th iterate of A006370).

0, 0, 0, 5, 2, 0, 0, 4, 0, 1, 2, 0, 7, 2, 0, 3, 4, 0, 0, 9, 0, 1, 2, 0, 0, 6, 0, 1, 2, 0, 5, 2, 0, 3, 4, 0, 0, 2, 0, 8, 2, 0, 0, 4, 0, 1, 7, 0, 5, 2, 0, 5, 6, 0, 0, 2, 0, 1, 2, 0, 92, 4, 0, 1, 2, 0, 7, 2, 0, 3, 9, 0, 0, 7, 0, 1, 2, 0, 0, 8, 0, 1, 2, 0, 5, 2, 0

Offset: 1

Rémy Sigrist, Dec 22 2022

nonn

If we travel the Collatz tree backwards, we will observe no further branching, whenever we have reached a number divisible by three. This is the reason why a(n) will be zero in all cases where n is divisible by three, because in this case no smaller number can exist further upward in the Collatz tree, as the actual branch will progress in numbers of the form 3*m*2^k. - Thomas Scheuerle, Dec 22 2022

			The Collatz sequence starting from 1 is: 1.
So a(1) = 0.
The Collatz sequence starting from 2 is: 2, 1.
So a(2) = 0.
The Collatz sequence starting from 3 is: 3, 10, 5, 16, 8, 4, 2, 1.
So a(3) = 0, a(10) = 1, a(5) = 2, a(16) = 3, a(8) = 4, a(4) = 5.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program
Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A006370, A070167, A177729.

nn = 87; c[] = -1; c[0] = 0; Do[(MapIndexed[If[c[#1] == -1, Set[c[#1], First[#2] - 1]] &, #]; -1 + Length[#]) &@ NestWhileList[If[EvenQ[#], #/2, 3 # + 1] &, n, c[#] == -1 && # > 1 &], {n, 0, nn}]; Array[c, nn] (* _Michael De Vlieger, Dec 23 2022 *)

PARI
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See Links section.
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a(n) = 0 iff n belongs to A177729.

If a(n) > 0 and n is odd, then a(n*3+1) - a(n) = 1. If a(n) > 0 and n is even, then a(n*3+1) - a(n*6+2) = 1. - Thomas Scheuerle, Dec 22 2022

A010120 Smallest start for a '3x+1' sequence containing 2^n.

Original entry on oeis.org

1, 2, 3, 3, 3, 21, 21, 75, 75, 151, 151, 1365, 1365, 5461, 5461, 14563, 14563, 87381, 87381, 184111, 184111, 932067, 932067, 5592405, 5592405, 13256071, 13256071, 26512143, 26512143, 357913941, 357913941, 1431655765, 1431655765, 3817748707

Offset: 0

Greg Warrington (waringtn(AT)netcom.com)

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

Cf. A054646.

a010120 = a070167 . a000079  -- Reinhard Zumkeller, Jan 02 2013

a(n) = A070167(A000079(n)). - Reinhard Zumkeller, Jan 02 2013

Corrected and extended by David W. Wilson, Jun 15 1997

A177729 Positive integers which do not appear in a Collatz sequence starting from a smaller positive integer.

Original entry on oeis.org

1, 2, 3, 6, 7, 9, 12, 15, 18, 19, 21, 24, 25, 27, 30, 33, 36, 37, 39, 42, 43, 45, 48, 51, 54, 55, 57, 60, 63, 66, 69, 72, 73, 75, 78, 79, 81, 84, 87, 90, 93, 96, 97, 99, 102, 105, 108, 109, 111, 114, 115, 117, 120, 123, 126, 127, 129, 132, 133, 135, 138, 141

Offset: 1

Raul D. Miller, May 12 2010

nonn

A variant of A061641, which is the main entry for this sequence.

The inclusion of 2 is apparently due to a non-standard definition of a Collatz sequence; A177729 assumes that the Collatz sequence ends when it reaches 1, whereas the standard definition includes the periodic 1,4,2,... from that point. The inclusion of 0 in A061641 is a bit odd, but is not actually wrong. One usually looks only at positive integers for Collatz sequences. - Franklin T. Adams-Watters, May 14 2010

			Collatz 1: 1; Collatz 2: 2,1; Collatz 3: 3,10,5,16,8,4,2,1; Collatz 6: 6,3,10,...

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
David Eisenbud and Brady Haran, UNCRACKABLE? The Collatz Conjecture, Numberphile Video, 2016.
Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A006577, A061641, A070167, A112695, A192719, A220263.

a177729 = head . a192719_row  -- Reinhard Zumkeller, Jan 03 2013

coll[n_]:=NestWhileList[If[EvenQ[#],#/2,3#+1]&,n,#>1&]; t={1}; Do[If[FreeQ[Union@@Table[coll[i],{i,n-1}],n],AppendTo[t,n]],{n,2,141}]; t (* Jayanta Basu, May 29 2013 *)

a(n) = A192719(n,1), see also A220263. - Reinhard Zumkeller, Jan 03 2013

A054646 Smallest number to give 2^(2n) in a hailstone (or 3x + 1) sequence.

Original entry on oeis.org

1, 3, 21, 75, 151, 1365, 5461, 14563, 87381, 184111, 932067, 5592405, 13256071, 26512143, 357913941, 1431655765, 3817748707, 22906492245, 91625968981, 244335917283, 1466015503701, 5212499568715, 10424999137431

Offset: 1

Jeff Heleen, Apr 16 2000

In hailstone sequences, only even powers of 2 are obtained as a final peak before descending to 1. [I assume this should really say: "These are numbers whose 3x+1 trajectory has the property that the final peak before descending to 1 is an even power of 2." - N. J. A. Sloane, Jul 22 2020]

For n>1, this a bisection of A010120. For n=3,6,7,9,12,15,16,18,19,21, we have a(n)=(4^n-1)/3, the largest possible value because one 3x+1 step produces 2^(2n). - T. D. Noe, Feb 19 2010

			The "3x+1" sequence starting at 21 is 21, 64, 32, 16, 8, 4, 2, 1, ..., and is the smallest start which contains 64 = 2^(2*3). So a(3) = 21. - _N. J. A. Sloane_, Jul 22 2020

J. Heleen, Final Peak Sequences for Hailstone Numbers, 1993, preprint. [Apparently unpublished as of June 2017]

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

a054646 1 = 1
a054646 n = a070167 $ a000302 n  -- Reinhard Zumkeller, Jan 02 2013

For n > 1: a(n) = A070167(A000302(n)). - Reinhard Zumkeller, Jan 02 2013

A192719 Chain of Collatz sequences.

Original entry on oeis.org

1, 2, 1, 3, 10, 5, 16, 8, 4, 2, 1, 6, 3, 10, 5, 16, 8, 4, 2, 1, 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1, 9, 28, 14, 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1, 12, 6, 3, 10, 5, 16, 8, 4, 2, 1

Offset: 1

Robert C. Lyons, Dec 31 2012

The sequence is a chain of Collatz sequences. The first Collatz sequence in the chain is (1). Each of the subsequent Collatz sequences in the chain starts with the minimum positive integer that does not appear in the previous Collatz sequences. If the Collatz conjecture is true, then each Collatz sequence in the chain will end with 1, and the chain will include an infinite number of distinct Collatz sequences. If the Collatz conjecture is false, then the chain will end with the first Collatz sequence that does not converge to 1.

T(n,1) = A177729(n). - Reinhard Zumkeller, Jan 03 2013

			The first Collatz sequence in the chain is (1). The second Collatz sequence in the chain is (2, 1), which starts with 2, since 2 is the smallest positive integer that doesn't appear the first Collatz sequence. The third Collatz sequence in the chain is (3, 10, 5, 16, 8, 4, 2, 1), which starts with 3, since 3 is the smallest positive integer that doesn't appear the previous Collatz sequences.
Thus this irregular array starts:
1;
2,  1;
3, 10,  5, 16,  8,  4,  2,  1;
6,  3, 10,  5, 16,  8,  4,  2,  1;
7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10,  5, 16,  8, 4,  2, 1;
9, 28, 14,  7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1;
...

Reinhard Zumkeller, Rows n = 1..120 of triangle, flattened
Robert C. Lyons, Chain of Collatz sequences generator program in Java
Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A014682, A070167, A070165, A061641, A177729, A192719.

Cf. A220263 (row lengths); A070165, A220237.

a192719 n k = a192719_tabf !! (n-1) !! (k-1)
a192719_row n = a192719_tabf !! (n-1)
a192719_tabf = f [1..] where
   f (x:xs) = (a070165_row x) : f (del xs $ a220237_row x)
   del us [] = us
   del us'@(u:us) vs'@(v:vs) | u > v     = del us' vs
                             | u < v     = u : del us vs'
                             | otherwise = del us vs
-- Reinhard Zumkeller, Jan 03 2013

Java
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See Lyons link.
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A101229 Perfect inverse "3x+1 conjecture" (See comments for rules).

Original entry on oeis.org

1, 2, 4, 1, 2, 4, 8, 16, 5, 10, 3, 6, 12, 24, 48, 96, 192, 384, 768, 1536, 3072, 6144, 12288, 24576, 49152, 98304, 196608, 393216, 786432, 1572864, 3145728, 6291456, 12582912, 25165824, 50331648, 100663296, 201326592, 402653184, 805306368

Offset: 1

Alexandre Wajnberg, Jan 22 2005

Perfect inverse "3x+1 conjecture": rule 1: multiply n by 2 to give n' = 2n. rule 2: when n'=(3x+1), do n"= (n'-1)/3 (n" integer) Additional rule: rule 2 is applied once for any number n' (otherwise, the sequence beginning with 1 would be the cycle "1 2 4 1 2 4 1 2 4 1..."); then apply rule 1.
This gives a particular sequence of hailstone numbers which may be considered as a central axis for all the hailstone number sequences. The perfect inverse "3x+1 conjecture" falls rapidly into the sequence 3 6 12 24 48 96... which will never give a number to which apply the 2nd rule.
a(n) for n >= 11 written in base 2: 11, 110, 11000, 110000, ..., i.e.: 2 times 1, (n-11) times 0 (see A003953(n-10)). - Jaroslav Krizek, Aug 17 2009

			The first 4 is followed by 1 because 4 = 3*1 + 1, so rule 2: (4-1)/3 = 1;
the second 4 is followed by 8 because the 2nd rule has already been applied, so rule 1: 4*2 = 8.

R. K. Guy, Collatz's Sequence, Section E16 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 215-218, 1994.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Collatz Problem
Index entries for linear recurrences with constant coefficients, signature (2).

Cf. A070165, A006577, A006667, A006666, A070167.

R:=PowerSeriesRing(Integers(), 45); Coefficients(R!( x*(17*x^10+27*x^8+7*x^3-1)/(2*x-1) )); // G. C. Greubel, Mar 20 2019

Rest[CoefficientList[Series[x*(17*x^10+27*x^8+7*x^3-1)/(2*x-1), {x, 0, 45}], x]] (* G. C. Greubel, Mar 20 2019 *)
LinearRecurrence[{2},{1,2,4,1,2,4,8,16,5,10,3},40] (* Harvey P. Dale, May 06 2023 *)

my(x='x+O('x^45)); Vec(x*(17*x^10+27*x^8+7*x^3-1)/(2*x-1)) \\ G. C. Greubel, Mar 20 2019

a=(x*(17*x^10+27*x^8+7*x^3-1)/(2*x-1)).series(x, 45).coefficients(x, sparse=False); a[1:] # G. C. Greubel, Mar 20 2019

a(n) = 3*2^(n-11) = 2^(n-11) + 2^(n-10) for n >= 11. - Jaroslav Krizek, Aug 17 2009
From Colin Barker, Apr 28 2013: (Start)
a(n) = 2*a(n-1) for n>11.
G.f.: x*(17*x^10+27*x^8+7*x^3-1) / (2*x-1). (End)

More terms from Joshua Zucker, May 18 2006

Edited by G. C. Greubel, Mar 20 2019

A358668 a(n) is the least m such that A359194^k(m) = n for some k >= 0 (where A359194^k denotes the k-th iterate of A359194).

Original entry on oeis.org

0, 0, 2, 3, 4, 5, 3, 7, 8, 9, 7, 11, 12, 3, 14, 11, 11, 17, 11, 19, 20, 14, 12, 23, 3, 12, 26, 12, 28, 29, 11, 12, 32, 33, 12, 35, 36, 11, 38, 12, 29, 41, 42, 28, 44, 45, 12, 47, 48, 26, 50, 51, 12, 53, 54, 3, 56, 26, 23, 59, 60, 12, 62, 26, 26, 65, 26, 67, 68

Offset: 0

Rémy Sigrist, Dec 22 2022

See A359214 for the corresponding minimal k's.

			The orbit of 0 under repeated application of A359194 is:
    0, 1, 0, ...
So a(0) = a(1) = 0.
The orbit of 2 under repeated application of A359194 is:
    2, 1, 0, 1, 0, ...
So a(2) = 2.
The orbit of 3 under repeated application of A359194 is:
    3, 6, 13, 24, 55, 90, 241, 300, 123, 142, 85, 0, 1, 0, ...
So a(3) = a(6) = a(13) = a(24) = a(55) = a(90) = a(241) = a(300) = a(123) = a(142) = a(85) = 3.

Rémy Sigrist, Table of n, a(n) for n = 0..10000
Rémy Sigrist, PARI program

Cf. A070167, A359194, A359214.

PARI
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See Links section.
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a(n) <= n.

A299962 Square array T(n, k) read by antidiagonals upwards, n > 0 and k > 0: T(n, k) is the k-th positive number whose Collatz sequence contains n.

Original entry on oeis.org

1, 2, 2, 3, 3, 3, 3, 6, 4, 4, 3, 4, 12, 5, 5, 6, 5, 5, 24, 6, 6, 7, 12, 6, 6, 48, 7, 7, 3, 9, 24, 7, 7, 96, 8, 8, 9, 5, 14, 48, 9, 8, 192, 9, 9, 3, 18, 6, 18, 96, 10, 9, 384, 10, 10, 7, 6, 36, 7, 28, 192, 11, 10, 768, 11, 11, 12, 9, 7, 72, 8, 36, 384, 12, 11

Offset: 1

Rémy Sigrist, Feb 22 2018

The n-th row corresponds to indices of rows in A070165 containing n.

			Array T(n, k) begins:
  n\k|  1     2     3     4     5     6     7     8     9    10
  ---+---------------------------------------------------------
    1|  1     2     3     4     5     6     7     8     9    10  -->  A000027 ?
    2|  2     3     4     5     6     7     8     9    10    11
    3|  3     6    12    24    48    96   192   384   768  1536  -->  A007283
    4|  3     4     5     6     7     8     9    10    11    12
    5|  3     5     6     7     9    10    11    12    13    14
    6|  6    12    24    48    96   192   384   768  1536  3072  -->  A091629
    7|  7     9    14    18    28    36    37    43    49    56
    8|  3     5     6     7     8     9    10    11    12    13
    9|  9    18    36    72   144   288   576  1152  2304  4608
   10|  3     6     7     9    10    11    12    13    14    15

Rémy Sigrist, PARI program for A299962
Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A000027, A007283, A091629, A070165, A070167.

PARI
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See Links section.
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T(n, 1) = A070167(n) for any n > 0.
T(3*n, k) = 3*n * 2^(k-1) for any n > 0 and k > 0.
If the Collatz conjecture is true, then:
- T(1, k) = k for any k > 0,
- T(2, k) = k+1 for any k > 0.

A372810 a(n) is the smallest number whose Collatz trajectory contains n, if trajectories do not terminate at 1 but continue to cycle through 1, 4, 2, 1, 4, 2, 1, ... .

Original entry on oeis.org

1, 1, 3, 1, 3, 6, 7, 3, 9, 3, 7, 12, 7, 9, 15, 3, 7, 18, 19, 7, 21, 7, 15, 24, 25, 7, 27, 9, 19, 30, 27, 21, 33, 7, 15, 36, 37, 25, 39, 7, 27, 42, 43, 19, 45, 15, 27, 48, 43, 33, 51, 7, 15, 54, 55, 37, 57, 19, 39, 60, 27, 27, 63, 21, 43, 66, 39, 45, 69, 15, 27

Offset: 1

Ethan E. Wood, May 13 2024

nonn

a(n) = A070167(n) for n >= 5.

a(n) = n if 3 divides n.

			For n=8,
  the trajectory of 1 is 1,  4, 2,  1, 4, ... (8 does not appear), and
  the trajectory of 2 is 2,  1, 4,  2, 1, ... (8 does not appear), but
  the trajectory of 3 is 3, 10, 5, 16, 8, ... (8 does appear),
so a(8) = 3.

R. K. Guy, Unsolved Problems in Number Theory, E16.

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

Cf. A070167 (sequence resulting if trajectories terminate at 1).

Edited by Jon E. Schoenfield, May 13 2024

cp's OEIS Frontend

A359226 a(n) is the least k >= 0 such that A006370^k(A070167(n)) = n (where A006370^k denotes the k-th iterate of A006370).

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A010120 Smallest start for a '3x+1' sequence containing 2^n.

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A177729 Positive integers which do not appear in a Collatz sequence starting from a smaller positive integer.

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A054646 Smallest number to give 2^(2n) in a hailstone (or 3x + 1) sequence.

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Haskell

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A192719 Chain of Collatz sequences.

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Java

A101229 Perfect inverse "3x+1 conjecture" (See comments for rules).

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Magma

Mathematica

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Sage

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A358668 a(n) is the least m such that A359194^k(m) = n for some k >= 0 (where A359194^k denotes the k-th iterate of A359194).

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