A006884 -id:A006884 - cp's OEIS frontend

A261690 a(1) = 1; for n>1, a(n) is the smallest number not already present which is entailed by the rules (i) k present => 3k+1 present; (ii) 2k present => k present.

1, 4, 2, 7, 13, 22, 11, 34, 17, 40, 20, 10, 5, 16, 8, 25, 31, 49, 52, 26, 61, 67, 76, 38, 19, 58, 29, 79, 88, 44, 94, 47, 103, 115, 121, 133, 142, 71, 148, 74, 37, 112, 56, 28, 14, 43, 85, 130, 65, 157, 169, 175, 184, 92, 46, 23, 70, 35, 106, 53, 139, 160, 80

Offset: 1

Vladimir Shevelev, Aug 28 2015

nonn

An analog of A109732 such that the statement 'the sequence is a permutation of the positive integers not divisible by 3' is equivalent to the (3*n+1)-conjecture for numbers not divisible by 3.
On Aug 29 2015, Max Alekseyev noted that, while the (3*n+1)-conjecture indeed implies that the sequence is a permutation of the positive integers not divisible by 3, the opposite statement is an open question. The author cannot yet prove this, so his previous comment is only a conjecture.
In connection with this, consider the following conjecture which could be called the (n-1)/3-conjecture. Let n be any number not divisible by 3. If n==1 (mod 3) and (n-1)/3 is not divisible by 3, then set n_1 = (n-1)/3. Otherwise set n_1 = 2*n. Conjecture. There exists an iteration n_m = 1. Does the (n-1)/3-conjecture imply the (3*n+1)-conjecture?
Example: 19->38->76->25->8->16->5->10->20->40->13->4->1.

Peter J. C. Moses, Table of n, a(n) for n = 1..10000

Cf. A006577, A006877, A006884, A109732.

A033958 In the '3x+1' problem, these values for the starting value set new records for number of steps to reach 1.

Original entry on oeis.org

1, 3, 7, 9, 25, 27, 73, 97, 129, 171, 231, 313, 327, 703, 871, 1161, 2463, 2919, 3711, 6171, 10971, 13255, 17647, 23529, 26623, 34239, 35655, 52527, 77031, 106239, 142587, 156159, 216367, 230631, 410011, 511935, 626331, 837799, 1117065, 1501353, 1723519, 2298025, 3064033

Offset: 1

N. J. A. Sloane

Only the 3x+1 steps, not the halving steps, are counted.

D. R. Hofstadter, Goedel, Escher, Bach: an Eternal Golden Braid, Random House, 1980, p. 400.
G. T. Leavens and M. Vermeulen, 3x+1 search problems, Computers and Mathematics with Applications, 24 (1992), 79-99.

Charles R Greathouse IV, Table of n, a(n) for n = 1..71
Brian Hayes, Computer Recreations: On the ups and downs of hailstone numbers, Scientific American, 250 (No. 1, 1984), pp. 10-16.
Index entries for sequences from "Goedel, Escher, Bach"
Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A006884, A006885, A006877, A006878, A033492, A033959.

a033958 n = a033958_list !! (n-1)
-- For definition of a033958_list: see A033959.
-- Reinhard Zumkeller, Jan 08 2014

f[ nn_ ] := Module[ {c, n}, c = 0; n = nn; While[ n != 1, If[ Mod[ n, 2 ] == 0, n /= 2, n = 3*n + 1; c++ ] ]; Return[ c ] ] maxx = -1; For[ n = 1, n <= 10^8, n++, Module[ {val}, val = f[ n ]; If[ val > maxx, maxx = val; Print[ n, " ", val ] ] ] ] (* Winston C. Yang (winston(AT)cs.wisc.edu), Aug 27 2000 *)

Positions of records in A006667. - Sean A. Irvine, Jul 22 2020

More terms from Jud McCranie, Jan 26 2000
Corrected with Mathematica code by Winston C. Yang (winston(AT)cs.wisc.edu), Aug 27 2000
a(40)-a(43) from Charles R Greathouse IV, Oct 07 2013

A075684 For odd numbers 2n-1, the maximum number produced by iterating the reduced Collatz function R defined as R(k) = (3k+1)/2^r, with r as large as possible.

Original entry on oeis.org

1, 5, 5, 17, 17, 17, 13, 53, 17, 29, 21, 53, 29, 3077, 29, 3077, 33, 53, 37, 101, 3077, 65, 45, 3077, 49, 77, 53, 3077, 65, 101, 61, 3077, 65, 101, 69, 3077, 3077, 113, 77, 269, 81, 3077, 85, 197, 101, 3077, 93, 3077, 3077, 149, 101, 3077, 269, 3077, 3077, 3077

Offset: 1

T. D. Noe, Sep 25 2002

See A075677 for the function R applied to the odd numbers once. See A075680 for the number of iterations required to yield 1. Sequence A006884, with the number 2 removed, gives the odd numbers that produce new record maxima. The maxima of the current sequence are related to A006885: if m is a maximum of the usual Collatz iteration, then (m-1)/3 is the maximum for the reduced Collatz iteration.

			a(4) = 17 because 7 is the fourth odd number and 17 is the largest number in the iteration: R(7)=11, R(11)=17, R(17)=13, R(13)=5, R(5)=1.

T. D. Noe, Table of n, a(n) for n = 1..5000

Cf. A006884, A006885, A075677, A075680.

nextOddK[n_] := Module[{m=3n+1}, While[EvenQ[m], m=m/2]; m]; (* assumes odd n *) Table[m=n; maxK=n; If[n>1, While[m=nextOddK[m]; maxK=Max[m, maxK]; m!=1]]; maxK, {n, 1, 200, 2}]

A095381 Initial values for 3x+1 trajectories in which the largest term arising in the iteration is a power of 2.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 10, 12, 16, 21, 32, 42, 64, 85, 128, 151, 170, 201, 227, 256, 302, 341, 402, 454, 512, 604, 682, 804, 908, 1024, 1365, 2048, 2730, 4096, 5461, 8192, 10922, 14563, 16384, 19417, 21845, 29126, 32768, 38834, 43690, 58252, 65536, 87381

Offset: 1

Labos Elemer, Jun 14 2004

nonn

Clearly the sequence is infinite and a(n) < 2^n. - Charles R Greathouse IV, May 25 2016

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A025586, A087256, A209229, A225124.

// Valid below A006884(47) = 12327829503 on 64-bit machines.
static long is (unsigned long n) {
  unsigned long r = n;
  n >>= __builtin_ctzl(n); // gcc builtin for A007814
  while (n > 1) {
    n = 3*n + 1;
    if (n > r) r = n;
    n >>= __builtin_ctzl(n);
  }
  return !(r & (r-1));
} // Charles R Greathouse IV, May 25 2016

a095381 n = a095381_list !! (n-1)
a095381_list = map (+ 1) $ elemIndices 1 $ map a209229 a025586_list
-- Reinhard Zumkeller, Apr 30 2013

Coll[n_]:=NestWhileList[If[EvenQ[#],#/2,3*#+1] &,n,#>1&];t={};Do[x = Max[Coll[n]];If[IntegerQ[Log[2,x]],AppendTo[t,n]],{n,90000}];t (* Jayanta Basu, Apr 28 2013 *)

is(n)=my(r=n); while(n>2, if(n%2, n=3*n+1; if(n>r, r=n)); n>>=1); r>>valuation(r,2)==1 \\ Charles R Greathouse IV, May 25 2016

A025586(a(n)) = 2^j for some j.

A095384 Number of different initial values for 3x+1 trajectories started with initial values not exceeding 2^n and in which the peak values are also not larger than 2^n.

Original entry on oeis.org

1, 2, 3, 4, 10, 13, 33, 55, 112, 181, 352, 580, 1072, 2127, 6792, 13067, 25906, 51447, 104575, 208149, 415921, 833109, 1661341, 3328124, 6648354, 13283680, 26533708, 53083687, 106166631, 212243709, 424564626, 848967377, 1698139390, 3396064464, 6791623786

Offset: 0

Labos Elemer, Jun 14 2004

nonn

			n=4: between iv={1,2,...,16} {2,8}U{3,5,6,10,12,16} provides peak values smaller than or equal with 16, so a(4) = 10 = A087256(4)+4

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

Cf. A087256, A095381, A095382, A095383.

Cf. A006884, A006885, A222292, A224538, A224540.

b:= proc(n) option remember; `if`(n=1, 1,
      max(n, b(`if`(n::even, n/2, 3*n+1))))
    end:
a:= proc(n) option remember; local t; t:=2^n;
      add(`if`(b(i)<=t, 1, 0), i=1..t)
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, Sep 26 2024

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}; Table[Print[{xm-1, m}];m=0; Do[If[ !Greater[Max[fpl[n]], 2^xm], m=m+1], {n, 1, 2^xm}], {xm, 1, 30}]
Collatz[n_] := NestWhileList[If[EvenQ[#], #/2, 3 # + 1] &, n, # > 1 &]; Table[Length[Select[Range[x=2^n], Max[Collatz[#]] <= x &]], {n,0,10}] (* T. D. Noe, Apr 29 2013 *)

a(21)-a(32) from Donovan Johnson, Feb 02 2011
a(0) from T. D. Noe, Apr 29 2013
a(33)-a(34) from Donovan Johnson, Jun 05 2013

A033959 Record number of steps to reach 1 in '3x+1' problem, corresponding to starting values in A033958.

Original entry on oeis.org

0, 2, 5, 6, 7, 41, 42, 43, 44, 45, 46, 47, 52, 62, 65, 66, 76, 79, 87, 96, 98, 101, 102, 103, 113, 114, 119, 125, 129, 130, 138, 141, 142, 164, 166, 174, 189, 195, 196, 197, 207, 208, 209, 217, 222, 228, 248, 256, 257, 258, 263, 278, 357, 358, 359, 362, 370

Offset: 1

N. J. A. Sloane

Only the 3x+1 steps, not the halving steps, are counted.

D. R. Hofstadter, Goedel, Escher, Bach: an Eternal Golden Braid, Random House, 1980, p. 400.
G. T. Leavens and M. Vermeulen, 3x+1 search problems, Computers and Mathematics with Applications, 24 (1992), 79-99.

Brian Hayes, Computer Recreations: On the ups and downs of hailstone numbers, Scientific American, 250 (No. 1, 1984), pp. 10-16.
Index entries for sequences from "Goedel, Escher, Bach"
Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A006884, A006885, A006877, A006878, A033492, A033958.

Cf. A075680.

a033959 n = a033959_list !! (n-1)
(a033959_list, a033958_list) = unzip $ (0, 1) : f 1 1 where
   f i x | y > x     = (y, 2 * i - 1) : f (i + 1) y
         | otherwise = f (i + 1) x
         where y = a075680 i
-- Reinhard Zumkeller, Jan 08 2014

A033959 := proc(n) local a,L; L := 0; a := n; while a <> 1 do if a mod 2 = 0 then a := a/2; else a := 3*a+1; L := L+1; fi; od: RETURN(L); end;

f[ nn_ ] := Module[ {c, n}, c = 0; n = nn; While[ n != 1, If[ Mod[ n, 2 ] == 0, n /= 2, n = 3*n + 1; c++ ] ]; Return[ c ] ] maxx = -1; For[ n = 1, n <= 10^8, n++, Module[ {val}, val = f[ n ]; If[ val > maxx, maxx = val; Print[ n, " ", val ] ] ] ]

More terms from Winston C. Yang (winston(AT)cs.wisc.edu), Aug 27 2000
More terms from Larry Reeves (larryr(AT)acm.org), Sep 27 2000
Offset corrected by Reinhard Zumkeller, Jan 08 2014

A025587 '3x+1' record-setters (blowup factor).

Original entry on oeis.org

1, 3, 7, 15, 27, 703, 1819, 4255, 4591, 9663, 26623, 60975, 77671, 113383, 159487, 1212415, 2684647, 3041127, 3873535, 4637979, 5656191, 6416623, 6631675, 19638399, 80049391, 210964383, 319804831, 1410123943, 70141259775, 77566362559

Offset: 0

David W. Wilson

This sequence uses the highest even number reached, which will always be a power of 2 larger than A295163. - Howard A. Landman, Nov 20 2017
A proper subsequence of A006884. - Robert G. Wilson v, Dec 23 2017
Let m be the maximum value in row n of A070165. This sequence is the record transform of the sequence m/n for n >= 1. - Michael De Vlieger, Mar 13 2018

Howard A. Landman, Table of n, a(n) for n = 0..33 (a(0)-a(23) from _David W. Wilson_, a(24)-a(26) from Larry Reeves, a(27) from _Jud McCranie_)
Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A295163 for maximum odd number reached, and A061523 for blowup factors.

Cf. A006884, A070165.

// First column is this sequence.
// Second column is the maximum (even) N reached.
// Third column is A061523, the ratio of those.
// NOTE: This could be made faster by special-casing 1,
// starting at 3, and incrementing by 4, since all terms except 1
// are congruent to 3 (mod 4).
#include    
long long    i=1, n, max_n;
long double    max_ratio=1.0, ratio;
int main()
{
    while(1)
    {
        n = i;
        max_n = n;
        while (n > i) // Can stop as soon as we drop below start.
        {
            n = 3*n + 1;
            max_n = (n > max_n) ? n : max_n;
            while (!(n&1))
            {
                n >>= 1;
            }
         }
        ratio = (double) max_n / (double) i;
        if (ratio > max_ratio)
        {
            max_ratio = ratio;
            printf("%lld\t%lld\t%Lf\n", i, max_n, max_ratio);
        }
        i += 2;
    }
}
// Howard A. Landman, Nov 14 2017

With[{s = Array[Max@ NestWhileList[If[EvenQ@#, #/2, 3 # + 1] &, #, # > 1 &]/# &, 2^18]}, Map[FirstPosition[s, #][[1]] &, Union@ FoldList[Max, s]]] (* Michael De Vlieger, Mar 13 2018 *)

More terms from Larry Reeves (larryr(AT)acm.org), May 03 2001
a(27) from Jud McCranie, Apr 23 2012
a(26) corrected (was missing least significant digit) by Howard A. Landman, Nov 14 2017

A222291 Least number whose Collatz (3x+1) trajectory has a number greater than 10^n.

Original entry on oeis.org

1, 3, 15, 27, 255, 703, 1819, 9663, 26623, 77671, 159487, 1212415, 4637979, 6631675, 19638399, 80049391, 319804831, 319804831, 319804831, 8528817511, 59436135663, 231913730799, 272025660543, 871673828443, 3716509988199, 3716509988199, 3716509988199

Offset: 0

T. D. Noe, Feb 19 2013

T. D. Noe, Table of n, a(n) for n = 0..37
Hisanori Mishima, Relation between n and max(n) (part of a chapter about the Collatz conjecture; errors in higher values)
Eric Roosendaal, 3x+1 path records
Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A006884 (3x+1 records), A222292 (base-2 version).

Cf. A224538.

A222292 Least number whose Collatz 3x+1 trajectory contains a number >= 2^n.

Original entry on oeis.org

1, 2, 3, 3, 3, 7, 15, 15, 27, 27, 27, 27, 27, 27, 447, 447, 703, 703, 1819, 1819, 1819, 4255, 4255, 9663, 9663, 20895, 26623, 60975, 60975, 60975, 77671, 113383, 159487, 159487, 159487, 665215, 1042431, 1212415, 2684647, 3041127, 4637979, 5656191, 6416623

Offset: 0

T. D. Noe, Feb 19 2013

Are the unique values a subset of A006884? - Ralf Stephan, May 27 2013

This sequence is important for the computation of Collatz numbers. It shows that using 32-bit integers, only numbers less than 159487 can have their Collatz trajectory computed.

T. D. Noe, Table of n, a(n) for n = 0..63
Eric Roosendaal, 3x+1 path records
Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A025586, A222291 (base-10 version).

Cf. A095384.

b:= proc(n) option remember; `if`(n=1, 1,
      max(n, b(`if`(n::even, n/2, 3*n+1))))
    end:
a:= proc(n) option remember; local i, t; t:=2^n;
      for i while b(i)Alois P. Heinz, Sep 25 2024

a(1) corrected by Kevin Ge, Sep 25 2024

A224538 Number of numbers k such that all terms of the Collatz (3x+1) iteration of k are <= 10^n.

Original entry on oeis.org

1, 4, 49, 340, 4235, 39706, 397068, 3970918, 39523168, 395436300, 3953296865

Offset: 0

T. D. Noe, Apr 24 2013

			For n = 1, the four k are 1, 2, 4, and 8.

Cf. A006884, A006885, A095384, A222291, A224540.

Collatz[n_] := NestWhileList[If[EvenQ[#], #/2, 3 # + 1] &, n, # > 1 &]; Table[Length[Select[Range[10^n], Max[Collatz[#]] <= 10^n &]], {n, 0, 5}]

a(10) from Donovan Johnson, Jun 05 2013

cp's OEIS Frontend

A261690 a(1) = 1; for n>1, a(n) is the smallest number not already present which is entailed by the rules (i) k present => 3*k+1 present; (ii) 2*k present => k present.

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A033958 In the '3x+1' problem, these values for the starting value set new records for number of steps to reach 1.

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A075684 For odd numbers 2n-1, the maximum number produced by iterating the reduced Collatz function R defined as R(k) = (3k+1)/2^r, with r as large as possible.

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Mathematica

A095381 Initial values for 3x+1 trajectories in which the largest term arising in the iteration is a power of 2.

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C

Haskell

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PARI

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A095384 Number of different initial values for 3x+1 trajectories started with initial values not exceeding 2^n and in which the peak values are also not larger than 2^n.

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Maple

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A033959 Record number of steps to reach 1 in '3x+1' problem, corresponding to starting values in A033958.

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Programs

Haskell

Maple

Mathematica

Extensions

A025587 '3x+1' record-setters (blowup factor).

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C

Mathematica

Extensions

A222291 Least number whose Collatz (3x+1) trajectory has a number greater than 10^n.

A261690 a(1) = 1; for n>1, a(n) is the smallest number not already present which is entailed by the rules (i) k present => 3k+1 present; (ii) 2k present => k present.