A033958 -id:A033958 - cp's OEIS frontend

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

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

A006667 Number of tripling steps to reach 1 from n in '3x+1' problem, or -1 if 1 is never reached.

Original entry on oeis.org

0, 0, 2, 0, 1, 2, 5, 0, 6, 1, 4, 2, 2, 5, 5, 0, 3, 6, 6, 1, 1, 4, 4, 2, 7, 2, 41, 5, 5, 5, 39, 0, 8, 3, 3, 6, 6, 6, 11, 1, 40, 1, 9, 4, 4, 4, 38, 2, 7, 7, 7, 2, 2, 41, 41, 5, 10, 5, 10, 5, 5, 39, 39, 0, 8, 8, 8, 3, 3, 3, 37, 6, 42, 6, 3, 6, 6, 11, 11, 1, 6, 40, 40, 1, 1, 9, 9, 4, 9, 4, 33, 4, 4, 38

Offset: 1

N. J. A. Sloane and Bill Gosper

A075680, which gives the values for odd n, isolates the essential behavior of this sequence. - T. D. Noe, Jun 01 2006

A033959 and A033958 give record values and where they occur. - Reinhard Zumkeller, Jan 08 2014

J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, p. 204, Problem 22.
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).

T. D. Noe, Table of n, a(n) for n = 1..10000
J. C. Lagarias, The 3x+1 problem and its generalizations, Amer. Math. Monthly, 92 (1985), 3-23.
Eric Weisstein's World of Mathematics, Collatz Problem.
Wikipedia, Collatz conjecture
Index entries for sequences related to 3x+1 (or Collatz) problem

Equals A078719(n)-1.

Cf. A000079, A000265, A006370, A006577, A006666 (halving steps), A092893, A127789, A139391, A209229.

a006667 = length . filter odd . takeWhile (> 2) . (iterate a006370)
a006667_list = map a006667 [1..]
-- Reinhard Zumkeller, Oct 08 2011

a:= proc(n) option remember; `if`(n<2, 0,
      `if`(n::even, a(n/2), 1+a(3*n+1)))
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Aug 08 2023

Table[Count[Differences[NestWhileList[If[EvenQ[#],#/2,3#+1]&,n,#>1&]], ?Positive], {n,100}] (* _Harvey P. Dale, Nov 14 2011 *)

for(n=2,100,s=n; t=0; while(s!=1,if(s%2==0,s=s/2,s=(3*s+1)/2; t++); if(s==1,print1(t,","); ); ))

def a(n):
    if n==1: return 0
    x=0
    while True:
        if n%2==0: n/=2
        else:
            n = 3*n + 1
            x+=1
        if n<2: break
    return x
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Apr 14 2017

a(1) = 0, a(n) = a(n/2) if n is even, a(n) = a(3n+1)+1 if n>1 is odd. The Collatz conjecture is that this defines a(n) for all n >= 1.
a(n) = A078719(n) - 1; a(A000079(n))=0; a(A062052(n))=1; a(A062053(n))=2; a(A062054(n))=3; a(A062055(n))=4; a(A062056(n))=5; a(A062057(n))=6; a(A062058(n))=7; a(A062059(n))=8; a(A062060(n))=9. - Reinhard Zumkeller, Oct 08 2011
a(n*2^k) = a(n), for all k >= 0. - L. Edson Jeffery, Aug 11 2014
a(n) = floor(log(2^A006666(n)/n)/log(3)). - Joe Slater, Aug 30 2017
a(n) = a(A085062(n)) + A007814(n+1) for n >= 2. - Alan Michael Gómez Calderón, Feb 07 2025
From  Alan Michael Gómez Calderón, Mar 31 2025: (Start)
a(n) = a(A139391(n)) + (n mod 2) for n >= 2;
a(n) = a(A139391(A000265(n))) - A209229(n) + 1 for n >= 2;
a(n) = a(A000265(A139391(n))) + (n mod 2) for n >= 2. (End)

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

"Escape clause" added to definition by N. J. A. Sloane, Jun 06 2017

A006878 Record number of steps to reach 1 in '3x+1' problem, corresponding to starting values in A006877.

Original entry on oeis.org

0, 1, 7, 8, 16, 19, 20, 23, 111, 112, 115, 118, 121, 124, 127, 130, 143, 144, 170, 178, 181, 182, 208, 216, 237, 261, 267, 275, 278, 281, 307, 310, 323, 339, 350, 353, 374, 382, 385, 442, 448, 469, 508, 524, 527, 530, 556, 559, 562, 583, 596, 612, 664, 685, 688, 691, 704

Offset: 1

N. J. A. Sloane, Robert Munafo

Both the 3x+1 steps and 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.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Hugo Pfoertner, Table of n, a(n) for n = 1..148 (from Eric Rosendaal's 3x+1 Delay Records, terms 1..130 from T. D. Noe)
Brian Hayes, Computer Recreations: On the ups and downs of hailstone numbers, Scientific American, 250 (No. 1, 1984), pp. 10-16.
J. C. Lagarias, The 3x+1 problem and its generalizations, Amer. Math. Monthly, 92 (1985), 3-23.
G. T. Leavens and M. Vermeulen, 3x+1 search programs, Computers and Mathematics with Applications, 24 (1992), 79-99. (Annotated scanned copy)
Eric Roosendaal, 3x+1 Delay Records
Robert G. Wilson v, Letter to N. J. A. Sloane with attachments, Jan. 1989
Index entries for sequences from "Goedel, Escher, Bach"
Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A006884, A006885, A006877, A033492, A033958, A033959.

f := 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; fi; L := L+1; od: RETURN(L); end;

numberOfSteps[x0_] := Block[{x = x0, nos = 0}, While[x != 1, If[Mod[x, 2] == 0, x = x/2, x = 3*x+1]; nos++]; nos]; A006878 = numberOfSteps /@ A006877 (* Jean-François Alcover, Feb 22 2012 *)
DeleteDuplicates[Table[Length[NestWhileList[If[EvenQ[#],#/2,3#+1]&,n,#>1&]],{n,0,10^6}],GreaterEqual]-1 (* The program generates the first 44 terms of the sequence, derived from all starting values from 1 up to and including 1 million. *) (* Harvey P. Dale, Nov 26 2022 *)

A112420 Leading digits in 3x+1 chain starting with the number 1117065.

Original entry on oeis.org

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

Offset: 1

Lekraj Beedassy, Dec 10 2005

The 3x+1 chain starts 1117065 -> 3351196 -> 1675598 -> 837799 and enters a 4->2->1->4->.. cycle after 527 iterations. The current sequence shows the most significant decimal digit after each iteration. [R. J. Mathar, Jul 26 2010]

Michel Marcus, Table of n, a(n) for n = 1..600
A. V. Kontorovich & S. J. Miller, Benford's Law, Values Of L-Functions And The 3x+1 Problem, arXiv:math/0412003 [math.NT], 2004-2005.
J. C. Lagarias and K. Soundararajan, Benford's Law for the 3x+1 Function, arXiv:math/0509175 [math.NT], 2005-2006.
P. Picart, Algorithme de Collatz et conjecture de Syracuse
Index entries for sequences related to Benford's law

Cf. A033958.

lista(nn) = my(c = 1117065); for (n=1, nn, print1(digits(c)[1], ", "); if (c % 2, c = 3*c+1, c = c/2);); \\ Michel Marcus, Nov 05 2016

a(103) corrected by Michel Marcus, Nov 05 2016

cp's OEIS Frontend

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

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A006667 Number of tripling steps to reach 1 from n in '3x+1' problem, or -1 if 1 is never reached.

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A006878 Record number of steps to reach 1 in '3x+1' problem, corresponding to starting values in A006877.

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Maple

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A112420 Leading digits in 3x+1 chain starting with the number 1117065.

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PARI

Extensions