A033958 -id:A033958

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A033959

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

+20
4

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

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

REFERENCES

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.

LINKS

Table of n, a(n) for n=1..57.

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

MAPLE

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;

MATHEMATICA

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 ] ] ] ]

PROG

(Haskell)

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

CROSSREFS

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

Cf. A075680.

KEYWORD

nonn,nice

AUTHOR

N. J. A. Sloane

EXTENSIONS

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

STATUS

approved

A006667

Number of tripling steps to reach 1 from n in '3x+1' problem, or -1 if 1 is never reached.
(Formerly M0019)

+10
59

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

COMMENTS

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

REFERENCES

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).

LINKS

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

FORMULA

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

MAPLE

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

MATHEMATICA

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

PROG

(PARI) 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, ", "); ); ))

(Haskell)

a006667 = length . filter odd . takeWhile (> 2) . (iterate a006370)

a006667_list = map a006667 [1..]

-- Reinhard Zumkeller, Oct 08 2011

(Python)

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

CROSSREFS

Equals A078719(n)-1.

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

KEYWORD

nonn,nice,hear,changed

AUTHOR

N. J. A. Sloane and Bill Gosper

EXTENSIONS

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

STATUS

approved

A006878

Record number of steps to reach 1 in '3x+1' problem, corresponding to starting values in A006877.
(Formerly M4335)

+10
11

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

COMMENTS

Both the 3x+1 steps and the halving steps are counted.

REFERENCES

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).

LINKS

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

MAPLE

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;

MATHEMATICA

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 *)

CROSSREFS

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

KEYWORD

nonn,nice

AUTHOR

N. J. A. Sloane, Robert Munafo

STATUS

approved

A112420

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

+10
2

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

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]

LINKS

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

PROG

(PARI) 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

CROSSREFS

Cf. A033958.

KEYWORD

nonn,base