A070165 -id:A070165 - cp's OEIS frontend

A225784 Denominators of the sum of the reciprocals of the Collatz (3x+1) sequence beginning at n.

1, 2, 240, 4, 80, 80, 272272, 8, 350064, 80, 38896, 240, 208, 272272, 4095840, 16, 3536, 116688, 21431696, 80, 1344, 38896, 1365280, 80, 535792400, 208, 44841486948146266934850832405421294927083491752830032389039800908293040266400, 38896, 1127984, 1365280

Offset: 1

Nico Brown, May 15 2013

nonn

If the sum of the reciprocals of a Collatz sequence is bounded, there are no Collatz cycles other than 4,2,1,4,2,1,...

a(n) = denominator of Sum_{k = 1..A006577(n)} 1/A070165(n,k). - Reinhard Zumkeller, May 16 2013

			For n=9 the Collatz sequence is {9, 28, 14, 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 4, 2, 1}.  So the sum of the reciprocals is 1/9 + 1/28 + 1/14 + 1/7 + 1/22 + 1/11 + ... + 1/4 + 1/2 + 1/1 = 1061683/350064, whose denominator is 350064.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A225761 (numerators), A087226.

Cf. A225843.

import Data.Ratio (denominator)
a225784 = denominator . sum . map (recip . fromIntegral) . a070165_row
-- Reinhard Zumkeller, May 16 2013

Collatz[n_] := NestWhileList[If[EvenQ[#], #/2, 3 # + 1] &, n, # > 1 &]; Table[Denominator[Total[1/Collatz[n]]], {n, 40}] (* T. D. Noe, May 15 2013 *)

Extended by T. D. Noe, May 15 2013

A347409 Longest run of halving steps in the trajectory from n to 1 in the Collatz map (or 3x+1 problem), or -1 if no such trajectory exists.

Original entry on oeis.org

0, 1, 4, 2, 4, 4, 4, 3, 4, 4, 4, 4, 4, 4, 5, 4, 4, 4, 4, 4, 6, 4, 5, 4, 4, 4, 5, 4, 4, 5, 5, 5, 4, 4, 5, 4, 4, 4, 4, 4, 5, 6, 4, 4, 4, 5, 5, 4, 4, 4, 4, 4, 5, 5, 5, 4, 4, 4, 4, 5, 5, 5, 5, 6, 4, 4, 4, 4, 4, 5, 5, 4, 5, 4, 8, 4, 4, 4, 4, 4, 5, 5, 5, 6, 8, 4, 4

Offset: 1

Paolo Xausa, Aug 30 2021

If the longest run of halving steps occurs as the final part of the trajectory, a(n) = A135282(n), otherwise a(n) > A135282(n).
Every nonnegative integer appears in the sequence at least once, since for any k >= 0 a(2^k) = k.
Conjecture: every integer >= 4 appears in the sequence infinitely many times.

			a(15) = 5 because the Collatz trajectory starting at 15 contains, as the longest halving run, a 5-step subtrajectory (namely, 160 -> 80 -> 40 -> 20 -> 10 -> 5).

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A006370, A070165, A135282.

Cf. A347668 (indices of records), A347669 (indices of first occurrences), A348007.

nterms=100;Table[c=n;sm=0;While[c>1,If[OddQ[c],c=3c+1,If[(s=IntegerExponent[c,2])>sm,sm=s];c/=2^s]];sm,{n,nterms}]

a(n)=my(nb=0); while (n != 1, if (n % 2, n=3*n+1, my(x = valuation(n, 2)); n /= 2^x; nb = max(nb, x));); nb; \\ Michel Marcus, Sep 03 2021

def A347409(n):
    m, r = n, 0
    while m > 1:
        if m % 2:
            m = 3*m + 1
        else:
            s = bin(m)[2:]
            c = len(s)-len(s.rstrip('0'))
            m //= 2**c
            r = max(r,c)
    return r # Chai Wah Wu, Sep 29 2021

A375266 Irregular triangle read by rows in which row n lists the iterates of the A375265 map from n to 1.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Aug 08 2024

By definition the trajectory ends when 1 is reached, so row 1 does not contain the cycle 1 -> 4 -> 2 -> 1.

See A375265 for links.

			Triangle begins:
   1;
   2,  1;
   3,  1;
   4,  2,  1;
   5, 16,  8,  4,  2,  1;
   6,  2,  1;
   7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10,  5, 16,  8,  4,  2,  1;
   8,  4,  2,  1;
   9,  3,  1;
  10,  5, 16,  8,  4,  2,  1;
  ...

Paolo Xausa, Table of n, a(n) for n = 1..12824 (rows 1..300 of the triangle, flattened).
Index entries for sequences related to 3x+1 (or Collatz) problem.

Cf. A375265, A375267 (# of iterations), A375268 (row sums), A375280 (row maxs).

Cf. A070165, A070168, A235795, A350279, A350548.

A375265[n_] := Which[Divisible[n, 3], n/3, Divisible[n, 2], n/2, True, 3*n + 1];
Array[NestWhileList[A375265, #, # > 1 &] &, 15]

A005184 Self-contained numbers: odd numbers k whose Collatz sequence contains a higher multiple of k.

Original entry on oeis.org

31, 83, 293, 347, 671, 19151, 2025797

Offset: 1

N. J. A. Sloane

The Collatz sequence of a number k is defined as a(1)=k, a(j+1) = a(j)/2 if a(j) is even, 3*a(j) + 1 if a(j) is odd.
No others less than 250000000. - Sam Handler (sam_5_5_5_0(AT)yahoo.com), Aug 07 2006
There are no more terms < 10^11. - Donovan Johnson, Nov 28 2013
There are no more terms < 10^15. - Alun Stokes, Mar 01 2021

			The Collatz sequence of 31 is 31, 94, 47, 142, 71, 214, 107, 322, 161, 484, 242, 121, 364, 182, 91, 274, 137, 412, 206, 103, 310 (see A008884) ... 310 is a multiple of 31, so the number 31 is "self-contained".

R. K. Guy, Unsolved Problems in Number Theory, E16.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alexander Rahn, Max Henkel, Sourangshu Ghosh, Eldar Sultanow, and Idriss Aberkane, An algorithm for linearizing Collatz convergence, hal-03286608 [math.DS], 2021.
Eldar Sultanow, Christian Koch, and Sean Cox, Collatz Sequences in the Light of Graph Theory, Universität Potsdam (Germany, 2020).
Index entries for sequences related to 3x+1 (or Collatz) problem

The ratios "higher multiple of k" / k are given in A059198.

Cf. A070165, A220237, A303698.

T:= proc(n) option remember; `if`(n=1, 1,
      [n, T(`if`(n::even, n/2, 3*n+1))][])
    end:
q:= n-> ormap(x-> x>n and irem(x, n)=0, [T(n)]):
select(q, [2*i-1$i=1..10000])[];  # Alois P. Heinz, Aug 04 2025

isSelfContained[n_] := Module[{d}, d = n; While[d != 1, If[EvenQ[d], d = d/2, d = 3 * d + 1]; If[IntegerQ[d/n], Return[True]]]; Return[False]]; For[n = 1, n <= 250000000, n += 2, If[isSelfContained[n], Print[n]]]; (* Sam Handler (sam_5_5_5_0(AT)yahoo.com), Aug 07 2006 *)
scnQ[n_] := MemberQ[Divisible[#, n] & / @Rest[NestWhileList[If[EvenQ[#], #/2, 3# + 1] &, n, # > 1 &]], True]; Select[Range[1, 2100001, 2], scnQ] (* Harvey P. Dale, Oct 21 2011 *)

m=5; d=2; while(1,n=(3*m+1)\2; until(n==1,n=if(n%2,3*n+1,n\2); if(n%m==0,print(m," ",n); break)); m+=d; d=6-d)

More terms from Robert G. Wilson v

Better description from Jack Brennen, Feb 07 2003

A177000 The Collatz iteration of these primes produces only even numbers, primes and 1.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 29, 37, 53, 59, 67, 89, 101, 131, 149, 157, 179, 181, 197, 241, 269, 277, 349, 397, 739, 853, 1109, 1237, 1429, 1621, 1861, 1877, 2161, 2389, 2531, 2957, 3413, 3797, 4549, 5717, 7621, 10069, 13397, 17749, 20021, 31541, 40277

Offset: 1

T. D. Noe, Apr 30 2010

The Collatz iteration of primes of the form (10*4^k-1)/3 produces only one additional prime: 5. The Collatz iteration of primes of the form (13*4^k-1)/3 produces only two additional primes: 5 and 13. This sequence is probably infinite.

In a sense, these are the simplest Collatz iterations starting with a prime number. Except for the increases (3x+1) when an odd prime occurs, the sequence produced by starting with a(n) is decreasing. All the primes that occur in such a Collatz iteration are in this sequence. - T. D. Noe, Oct 05 2011

			The Collatz iteration of 7 produces 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, and 1, which are either even, prime, or 1.

Donovan Johnson and T. D. Noe, Table of n, a(n) for n = 1..10000 (Donovan Johnson to 203 terms)
Eric Weisstein's World of Mathematics, Collatz Problem
Wikipedia, Collatz conjecture
Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A055509, A078350, A078373, A070165, A010051.

a177000 n = a177000_list !! (n-1)
a177000_list = filter (all (\x -> even x || a010051' x == 1) .
                       (init . a070165_row)) a000040_list
-- Reinhard Zumkeller, Apr 03 2012

Collatz[n_] := NestWhileList[If[EvenQ[ # ], #/2, 3#+1] &, n, #>1 &]; Reap[Do[p=Prime[n]; s=Most[Select[Collatz[p],OddQ]]; If[And@@PrimeQ[s], Sow[p]], {n,PrimePi[10^4]}]][[2,1]]
oenQ[n_]:=AllTrue[DeleteCases[Most[NestWhileList[If[EvenQ[#],#/2, 3#+1]&,n, #>1&]], ?PrimeQ],EvenQ]; Select[Prime[Range[5000]], oenQ] (* The program uses the AllTrue function from Mathematica version 10 *) (* _Harvey P. Dale, Dec 28 2014 *)

is(n)=isprime(n) && (n<23 || is((3*n+1)>>valuation(3*n+1,2))) \\ Charles R Greathouse IV, Jun 20 2013

A178168 Product of the numbers in the Collatz (3x+1) trajectory of n, including n.

Original entry on oeis.org

1, 2, 153600, 8, 5120, 921600, 704889816350720000, 64, 2486851272085340160000, 51200, 4577206599680000, 11059200, 532480000, 9868457428910080000, 114523513552896000000000, 1024, 12238520320000, 44763322897536122880000

Offset: 1

T. D. Noe, May 21 2010

nonn

Row n of A070165 has the Collatz trajectory of n. It appears that all products are unique. This has been verified for all n < 10^6 and for the 12332052 values of n for which a(n) < 2^1081.

			The Collatz iteration starting with 3 is 3, 10, 5, 16, 8, 4, 2, 1. The product of these numbers is 153600.

T. D. Noe, Table of n, a(n) for n=1..500

Cf. A087226 (LCM of the trajectory), A178169, A178170

Collatz[n_] := NestWhileList[If[EvenQ[ # ], #/2, 3#+1] &, n, #>1 &]; Table[Times@@Collatz[n], {n, 100}]

A207674 Numbers such that all divisors occur in their Collatz trajectories.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 16, 17, 19, 20, 22, 23, 24, 25, 26, 28, 29, 31, 32, 34, 37, 38, 40, 41, 43, 44, 46, 47, 48, 49, 50, 52, 53, 56, 58, 59, 61, 62, 64, 65, 67, 68, 71, 73, 74, 76, 79, 80, 82, 83, 86, 88, 89, 92, 94, 97, 98, 100, 101

Offset: 1

Reinhard Zumkeller, Feb 20 2012

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Collatz Problem
Wikipedia, Collatz conjecture
Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A027750, A070165, A006370, A207675 (complement), A000079 and A000040 are subsequences.

import Data.List (intersect)
a207674 n = a207674_list !! (n-1)
a207674_list = filter
   (\x -> a027750_row x `intersect` a070165_row x == a027750_row x) [1..]

coll[n_]:=NestWhileList[If[EvenQ[#],#/2,3#+1]&,n,#>1&]; Select[Range[101],Complement[Divisors[#],coll[#]]=={}&] (* Jayanta Basu, May 27 2013 *)

A220145 The Collatz (3x+1) iteration mod 2 with bits combined.

Original entry on oeis.org

1, 10, 10000101, 100, 100001, 100001010, 10000100010010101, 1000, 10000100010010101001, 1000010, 100001000100101, 1000010100, 1000010001, 100001000100101010, 100001000001010101, 10000, 1000010001001, 100001000100101010010, 100001000100101000101, 10000100

Offset: 1

T. D. Noe, Jan 17 2013

This is essentially sequence A070165 mod 2 with bits in the same iteration combined. Note that A176999 is similar, but with a different encoding.

It appears that all numbers are distinct. Sequence A005186 tells how many numbers produce bit strings of a given length. Sequence A221468 converts to decimal and A221467 sorts them.

			For n = 7, the Collatz iteration is 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1. Looking at these numbers in base 2 and reversing them, we obtain 10000100010010101.

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A005186, A070165, A176999, A221467, A221468.

Collatz[n_] := NestWhileList[If[EvenQ[#], #/2, 3 # + 1] &, n, # > 1 &]; Table[FromDigits[Mod[Reverse[Collatz[n]], 2]], {n, 30}]

A221213 Conjectured total number of times that k-n appears in the Collatz (3x+1) sequence of k for k = 1, 2, 3,....

Original entry on oeis.org

42, 42, 57, 52, 46, 53, 58, 57, 54, 49, 56, 59, 49, 62, 61, 59, 69, 59, 50, 54, 72, 64, 65, 55, 57, 54, 55, 66, 61, 60, 62, 61, 64, 73, 62, 59, 71, 63, 62, 58, 68, 72, 63, 59, 57, 65, 70, 59, 60, 59, 71, 55, 64, 54, 66, 75, 67, 62, 64, 64, 73, 68, 68, 67, 58, 61

Offset: 1

Jayanta Basu, Feb 21 2013

nonn

Values are tested for natural numbers up to 1000000.

			a(1) = 42 = total number of k such that k-1 appears in the Collatz sequence of k, that is, the number of terms in A070991.

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

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A070165, A070991.

Collatz[n_] := NestWhileList[If[EvenQ[#], #/2, 3 # + 1] &, n, # > 1 &]; nn = 75; t = Table[0, {nn}]; lastChange = 10; k = 0; While[k < 2*lastChange, k++; c = Collatz[k]; d = Intersection[Range[nn], k - c]; If[Length[d] > 0, lastChange = k; t[[d]]++]]; t (* T. D. Noe, Feb 21 2013 *)

A221469 Number of increasing peak values in the Collatz (3x+1) iteration of n.

Original entry on oeis.org

0, 0, 2, 0, 1, 2, 3, 0, 3, 1, 2, 1, 1, 3, 4, 0, 1, 3, 2, 0, 1, 2, 3, 0, 2, 1, 15, 2, 1, 4, 14, 0, 1, 1, 2, 1, 1, 2, 4, 0, 14, 1, 2, 1, 1, 3, 13, 0, 1, 2, 2, 0, 1, 15, 13, 0, 2, 1, 3, 3, 1, 14, 12, 0, 1, 1, 2, 0, 1, 2, 12, 0, 13, 1, 2, 1, 1, 4, 4, 0, 1, 14, 12

Offset: 1

T. D. Noe, Jan 17 2013

nonn

That is, the number of times that the Collatz iteration of n reaches a new maximum. See A221470 for the first occurrence of each peak count.

			The Collatz iteration starting at 7 is (7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1), which has 3 increasing peaks: 22, 34, and 52.

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A070165 (Collatz trajectory of n), A221470.

a221469 n = sum $ map fromEnum $ zipWith (>) (tail ts) ts where
   ts = scanl1 max $ a070165_row n
-- Reinhard Zumkeller, Jan 18 2013

Collatz[n_] := NestWhileList[If[EvenQ[#], #/2, 3 # + 1] &, n, # > 1 &]; Table[c = Collatz[n]; cnt = 0; mx = n; Do[If[k > mx, cnt++; mx = k], {k, c}]; cnt, {n, 100}]

cp's OEIS Frontend

A225784 Denominators of the sum of the reciprocals of the Collatz (3x+1) sequence beginning at n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

Extensions

A347409 Longest run of halving steps in the trajectory from n to 1 in the Collatz map (or 3x+1 problem), or -1 if no such trajectory exists.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

A375266 Irregular triangle read by rows in which row n lists the iterates of the A375265 map from n to 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A005184 Self-contained numbers: odd numbers k whose Collatz sequence contains a higher multiple of k.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Extensions

A177000 The Collatz iteration of these primes produces only even numbers, primes and 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

A178168 Product of the numbers in the Collatz (3x+1) trajectory of n, including n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A207674 Numbers such that all divisors occur in their Collatz trajectories.

Views

Author

Keywords

Links

Crossrefs

Programs

Haskell

Mathematica