A092892 -id:A092892 - cp's OEIS frontend

A138895 Triangle read by rows: T(n,k) = A092892(n-k) for 0 <= k <= n.

1, 2, 1, 4, 2, 1, 8, 4, 2, 1, 5, 8, 4, 2, 1, 3, 5, 8, 4, 2, 1, 6, 3, 5, 8, 4, 2, 1, 12, 6, 3, 5, 8, 4, 2, 1, 24, 12, 6, 3, 5, 8, 4, 2, 1, 17, 24, 12, 6, 3, 5, 8, 4, 2, 1, 11, 17, 24, 12, 6, 3, 5, 8, 4, 2, 1

Offset: 0

Paul Barry, Apr 02 2008

Matrix in Toeplitz form related to the Collatz (3x + 1) algorithm.

Row sums are A138847. Antidiagonal sums are A138848.

			Triangle begins
   1;
   2,  1;
   4,  2,  1;
   8,  4,  2,  1;
   5,  8,  4,  2,  1;
   3,  5,  8,  4,  2,  1;
   6,  3,  5,  8,  4,  2,  1;
  12,  6,  3,  5,  8,  4,  2,  1;
  24, 12,  6,  3,  5,  8,  4,  2,  1;
  17, 24, 12,  6,  3,  5,  8,  4,  2,  1;
  11, 17, 24, 12,  6,  3,  5,  8,  4,  2,  1;
  ...

Cf. A092892 (1st column), A138847, A138848.

Edited by Georg Fischer, Jul 28 2023

A006666 Number of halving steps to reach 1 in '3x+1' problem, or -1 if this never happens.

Original entry on oeis.org

0, 1, 5, 2, 4, 6, 11, 3, 13, 5, 10, 7, 7, 12, 12, 4, 9, 14, 14, 6, 6, 11, 11, 8, 16, 8, 70, 13, 13, 13, 67, 5, 18, 10, 10, 15, 15, 15, 23, 7, 69, 7, 20, 12, 12, 12, 66, 9, 17, 17, 17, 9, 9, 71, 71, 14, 22, 14, 22, 14, 14, 68, 68, 6, 19, 19, 19, 11, 11, 11, 65, 16, 73, 16, 11, 16

Offset: 1

N. J. A. Sloane, Bill Gosper

Equals the total number of steps to reach 1 under the modified '3x+1' map: T(n) = n/2 if n is even, (3n+1)/2 if n is odd (see A014682).

Pairs of consecutive integers of the same height occur infinitely often and in infinitely many different patterns (Garner 1985). - Joe Slater, May 24 2018

			2 -> 1 so a(2) = 1; 3 -> 10 -> 5 -> 16 -> 8 -> 4 -> 2 -> 1, with 5 halving steps, so a(3) = 5; 4 -> 2 -> 1 has two halving steps, so a(4) = 2; etc.

R. K. Guy, Unsolved Problems in Number Theory, E16.
J. C. Lagarias, ed., The Ultimate Challenge: The 3x+1 Problem, Amer. Math. Soc., 2010.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..10000
David Eisenbud and Brady Haran, UNCRACKABLE? The Collatz Conjecture, Numberphile Video, 2016.
Lynn E. Garner, On Heights in the Collatz 3n+1 Problem, Discrete Math. 55 (1985) 57-64.
J. C. Lagarias, The 3x+1 problem and its generalizations, Amer. Math. Monthly, m92 (1985), 3-23.
Jeffrey C. Lagarias, The 3x+1 Problem: An Overview, arXiv:2111.02635 [math.NT], 2021.
K. Matthews, The Collatz Conjecture
Eric Weisstein's World of Mathematics, Collatz Problem
Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A006370, A006577, A006667 (tripling steps), A014682, A092892, A127789 (record indices of 2^a(n)/(3^A006667(n)*n)).

a006666 = length . filter even . takeWhile (> 1) . (iterate a006370)
-- Reinhard Zumkeller, Oct 08 2011

# A014682
T:=proc(n) if n mod 2 = 0 then n/2 else (3*n+1)/2; fi; end;
# A006666
t1:=[0]:
for n from 2 to 100 do
L:=1; p := n;
while T(p) <> 1 do p:=T(p); L:=L+1; od:
t1:=[op(t1),L];
od: t1;

Table[Count[NestWhileList[If[OddQ[#],3#+1,#/2]&,n,#>1&],?(EvenQ[#]&)], {n,80}] (* _Harvey P. Dale, Sep 30 2011 *)

a(n)=my(t); while(n>1, if(n%2, n=3*n+1, n>>=1; t++)); t \\ Charles R Greathouse IV, Jun 21 2017

def a(n):
    if n==1: return 0
    x=0
    while True:
        if not n%2:
            n//=2
            x+=1
        else: n = 3*n + 1
        if n<2: break
    return x
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Apr 14 2017

A092892(a(n)) = n and A092892(m) <> n for m < a(n). - Reinhard Zumkeller, Mar 14 2014
a(2^n) = n. - Bob Selcoe, Apr 16 2015
a(n) = ceiling(log(n*3^A006667(n))/log(2)). - Joe Slater, Aug 30 2017
a(2^k-1) = a(2^(k+1)-1)-1, for odd k>1. - Joe Slater, May 17 2018
a(n) = a(A085062(n)) + A007814(n+1) + 1 for n >= 2. - Alan Michael Gómez Calderón, Feb 01 2025

More terms from Larry Reeves (larryr(AT)acm.org), Apr 27 2001

Name edited by M. F. Hasler, May 07 2018

A092893 Smallest starting value in a Collatz '3x+1' sequence such that the sequence contains exactly n tripling steps.

Original entry on oeis.org

1, 5, 3, 17, 11, 7, 9, 25, 33, 43, 57, 39, 105, 135, 185, 123, 169, 219, 159, 379, 283, 377, 251, 167, 111, 297, 395, 263, 175, 233, 155, 103, 137, 91, 121, 161, 107, 71, 47, 31, 41, 27, 73, 97, 129, 171, 231, 313, 411, 543, 731, 487, 327, 859, 1145, 763, 1017, 1351

Offset: 0

Hugo Pfoertner, Mar 11 2004

nonn

First occurrence of n in A006667.
These are the odd (primitive) terms in A129304. - T. D. Noe, Apr 09 2007
Except for a(1) = 5, all values are congruent {1, 3, 7} (mod 8). Reason: If n is 5 (mod 8) then the Collatz trajectory starting with m = (n - 1)/4 contains the same number of tripling steps, because n = 4m + 1 and the Collatz 3x + 1 step results in 3*(4m + 1) + 1 = 12m + 4 which gets reduced by halving to 3m + 1, without changing the number of tripling steps. - Ralf Stephan, Jun 19 2025

			a(4)=11 because the Collatz sequence 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1 is the first sequence containing 4 tripling steps.

T. D. Noe, Table of n, a(n) for n=0..300
Jeffrey R. Goodwin, The 3x+1 Problem and Integer Representations, Arxiv preprint arXiv:1504.03040 [math.NT], 2015.
Eric Weisstein's World of Mathematics, Collatz Problem
Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A006667, A033491, A092892, A087228.

Row n=1 of A354236.

a[n_]:=Length[Select[NestWhileList[If[EvenQ[#],#/2,3#+1] &,n,#>1 &],OddQ]]; Table[i=1; While[a[i]!=n,i=i+2]; i,{n,58}] (* Jayanta Basu, May 27 2013 *)

A138847 Row sums of a Collatz triangle.

Original entry on oeis.org

1, 3, 7, 15, 20, 23, 29, 41, 65, 82, 93, 100, 114, 123, 141, 177, 202, 251, 284, 349, 392, 478, 535, 574, 652, 805, 910, 1113, 1248, 1518, 1703, 1826, 2072, 2241, 2570, 2789, 2948, 3243, 3812, 4191, 4474, 4979, 5356, 5607, 5774, 5885, 6107

Offset: 0

Paul Barry, Mar 31 2008, Apr 02 2008

Partial sums of A092892.

Row sums of A138895.

a(n) = Sum_{k=0..n} A092892(k).

a(n) = Sum_{k=0..n} A138895(n,k).

A138848 Diagonal sums of a Collatz triangle.

Original entry on oeis.org

1, 2, 5, 10, 10, 13, 16, 25, 40, 42, 51, 49, 65, 58, 83, 94, 108, 143, 141, 208, 184, 294, 241, 333, 319, 486, 424, 689, 559, 959, 744, 1082, 990, 1251, 1319, 1470, 1478, 1765, 2047, 2144, 2330, 2649, 2707, 2900, 2874, 3011, 3096, 3455

Offset: 0

Paul Barry, Mar 31 2008, Apr 02 2008

Diagonal sums of A138895.

a(n)=sum{k=0..floor(n/2), A092892(n-2k)}

A138846 Erroneous version of A138895.

Original entry on oeis.org

1, 2, 1, 4, 2, 1, 8, 4, 2, 1, 5, 8, 4, 2, 1, 3, 5, 8, 4, 2, 1, 6, 3, 5, 8, 4, 2, 1, 12, 6, 3, 5, 8, 4, 2, 1, 24, 12, 6, 3, 5, 8, 4, 2, 1, 17, 26, 13, 20, 10, 5, 8, 4, 2, 1, 11, 17, 26, 13, 20, 10, 5, 8, 4, 2, 1

Offset: 0

Paul Barry, Mar 31 2008, Apr 02 2008

dead

Former definition was: Triangle read by rows: T(n,k) = A092892(n-k) if k <= n, 0 otherwise.

			Triangle begins
   1;
   2,  1;
   4,  2,  1;
   8,  4,  2,  1;
   5,  8,  4,  2,  1;
   3,  5,  8,  4,  2,  1;
   6,  3,  5,  8,  4,  2,  1;
  12,  6,  3,  5,  8,  4,  2,  1;
  24, 12,  6,  3,  5,  8,  4,  2,  1;
  17, 26, 13, 20, 10,  5,  8,  4,  2,  1;
  11, 17, 26, 13, 20, 10,  5,  8,  4,  2,  1;

cp's OEIS Frontend

A138895 Triangle read by rows: T(n,k) = A092892(n-k) for 0 <= k <= n.

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A006666 Number of halving steps to reach 1 in '3x+1' problem, or -1 if this never happens.

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A092893 Smallest starting value in a Collatz '3x+1' sequence such that the sequence contains exactly n tripling steps.

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A138847 Row sums of a Collatz triangle.

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A138848 Diagonal sums of a Collatz triangle.

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A138846 Erroneous version of A138895.

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