A074472 -id:A074472 - cp's OEIS frontend

A075487 Length of iteration list when Collatz-function is iterated with initial value 1+3^n.

2, 3, 7, 19, 111, 22, 96, 33, 76, 75, 43, 135, 134, 133, 132, 144, 205, 129, 190, 140, 95, 94, 261, 428, 91, 258, 394, 331, 255, 254, 390, 389, 388, 462, 461, 460, 459, 458, 457, 456, 455, 454, 453, 501, 500, 450, 498, 497, 752, 495, 494, 493, 748, 491, 746, 489

Offset: 0

Labos Elemer, Sep 26 2002

nonn

n=2, 1+3^n = 10, list = {10,5,16,8,4,2,1}, so a(2)=7

T. D. Noe, Table of n, a(n) for n = 0..1000

Cf. A006370, A008908, A074472, A075484-A075488.

coll[n_]:=Length[NestWhileList[If[EvenQ[#],#/2,3#+1]&,n,#!=1&]]; coll /@ (3^Range[0, 60] + 1) (* Harvey P. Dale, Dec 15 2014 *)

a(n) = A008908(3^n+1).

A074474 Number of iteration that first becomes smaller than the initial value if Collatz-function (A006370) is iterated, starting with numbers of form 4n+3.

Original entry on oeis.org

7, 12, 9, 12, 7, 9, 97, 92, 7, 14, 9, 89, 7, 9, 12, 89, 7, 84, 9, 14, 7, 9, 74, 14, 7, 69, 9, 51, 7, 9, 14, 25, 7, 12, 9, 12, 7, 9, 66, 35, 7, 48, 9, 14, 7, 9, 12, 22, 7, 14, 9, 22, 7, 9, 14, 51, 7, 20, 9, 33, 7, 9, 45, 22, 7, 12, 9, 12, 7, 9, 40, 17, 7, 14, 9, 22, 7, 9, 12, 35, 7, 35, 9, 14

Offset: 1

Labos Elemer, Sep 19 2002

nonn

			n=6: 4n-1=23, the list={23,70,35,106,53,160,80,40,20, 10,5,16,8,4,2,1} sinks first below iv=23 at 20, the 9th term, so a(6)=9. Observe several (provable) modular rules with respect of initial value: e.g. regular appearance of 9 and 7.

Cf. A006370, A074472, A074473.

Table[Function[k, Length@ NestWhileList[If[EvenQ@ #, #/2, 3 # + 1] &, k, # >= k &]][4 n - 1], {n, 120}] (* Michael De Vlieger, Feb 20 2017 *)

a(n)=A074473[4n-1]

A075483 Number of iteration that first becomes smaller than the initial value if Collatz-function (A006370) is iterated, starting with numbers of form 64n+63.

Original entry on oeis.org

89, 25, 22, 22, 35, 20, 66, 30, 17, 38, 133, 27, 25, 40, 35, 30, 20, 25, 22, 38, 38, 133, 51, 27, 17, 40, 22, 30, 20, 35, 22, 95, 131, 33, 20, 25, 27, 22, 27, 66, 17, 27, 71, 45, 33, 48, 35, 89, 22, 33, 30, 30, 48, 22, 40, 30, 17, 61, 30, 64, 22, 22, 25, 84, 22, 22, 25, 33

Offset: 0

Labos Elemer, Sep 24 2002

nonn

1stSubmergeLengths[=A074473] with initial values belonging to other residue classes modulo 64 are either listed in A075476-A075483 or can be easily determined. For 64k+2s the first sink below initial value is at 2nd iterate; for 64k+4s+1 the first submerge below initial value comes at 4th term of iteration list; finally if initial value is of 64k+4s+3 form or moreover initial value = 64k+r, r = 3, 11, 19, 23, 35, 43, 51, 55, then for all k first sink emerges at the 7th, 9th, 7th, 9th, 7th, 9th, 7th, 9th iterates, respectively.

			n=8: 64n + 63 = 575, the list = the 17th term 410 < 575 = initial value, so a(8)=17.

Cf. A006370, A074472, A074473, A075476-A075482.

Table[Function[k, Length@ NestWhileList[If[EvenQ@ #, #/2, 3 # + 1] &, k, # >= k &]][64 n - 1], {n, 120}] (* Michael De Vlieger, Feb 20 2017 *)

a(n) = A074473(64n + 63).

A075488 Length of iteration list when Collatz-function is iterated with initial value -1+3^n.

Original entry on oeis.org

2, 4, 11, 10, 97, 96, 33, 32, 44, 43, 135, 134, 133, 132, 100, 99, 191, 190, 140, 139, 262, 261, 428, 427, 395, 394, 331, 330, 391, 390, 389, 388, 462, 461, 460, 459, 458, 457, 456, 455, 454, 453, 501, 500, 499, 498, 497, 496, 495, 494, 493, 492, 747, 746

Offset: 1

Labos Elemer, Sep 26 2002

nonn

n=3, -1+3^n = 26, list = {26,13,40,20,10,5,16,8,4,2,1}, a(3)=11.

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A006370, A008908, A074472, A075484-A075487.

lcoll[n_] := Length[NestWhileList[If[EvenQ[#], #/2, 3 # + 1] &, n, # > 1 &]]; Table[lcoll[3^i - 1], {i, 54}] (* Jayanta Basu, Jun 15 2013 *)

a(n) = A008908(3^n-1).

A075482 Number of iteration that first becomes smaller than the initial value if Collatz-function (A006370) is iterated, starting with numbers of the form 64n + 59.

Original entry on oeis.org

12, 14, 12, 45, 12, 14, 12, 17, 12, 14, 12, 33, 12, 14, 12, 20, 12, 14, 12, 25, 12, 14, 12, 17, 12, 14, 12, 20, 12, 14, 12, 30, 12, 14, 12, 25, 12, 14, 12, 17, 12, 14, 12, 30, 12, 14, 12, 22, 12, 14, 12, 69, 12, 14, 12, 17, 12, 14, 12, 22, 12, 14, 12, 22, 12, 14, 12, 82, 12

Offset: 0

Labos Elemer, Sep 24 2002

nonn

1stSubmergeLengths[=A074473] with initial values belonging to other residue classes modulo 64 are either listed in A075476-A075483 or can be easily determined. For 64k+2s the first sink below initial value is at 2nd iterate; for 64k+4s+1 the first submerge below initial value comes at 4th term of iteration list; finally if initial value is of 64k+4s+3 form or moreover initial value = 64k+r, r = 3, 11, 19, 23, 35, 43, 51, 55, then for all k first sink emerges at the 7th, 9th, 7th, 9th, 7th, 9th, 7th, 9th iterates, respectively.

			n=0: 64n + 59 = 59, the list = {59,178,89,268,134,67,202,101,304,152,76,38,...} the 12th term = 38 < 59 = the initial value, so a(0)=12.

Antti Karttunen, Table of n, a(n) for n = 0..16384
Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A006370, A074472, A074473, A075476, A075477-A075483.

Table[Function[m, Length@ NestWhileList[If[EvenQ@ #, #/2, 3 # + 1] &, m, # >= m - 1 &]][64 n + 59], {n, 0, 84}] (* Michael De Vlieger, Mar 25 2017 *)

A006370(n) = if(n%2, 3*n+1, n/2);
A074473(n) = if(1==n,n,my(org_n=n); for(i=1,oo,if(nA006370(n)));
A075482(n) = A074473((64*n)+59); \\ Antti Karttunen, Oct 09 2018

a(n) = A074473(64n + 59).

A075484 Length of iteration-list when Collatz-function(A006370) is iterated with initial value 5^n.

Original entry on oeis.org

1, 6, 24, 109, 26, 124, 147, 139, 100, 92, 115, 337, 135, 277, 181, 261, 240, 219, 286, 322, 451, 337, 303, 432, 243, 540, 408, 444, 304, 464, 438, 554, 484, 582, 517, 677, 462, 617, 1002, 539, 655, 709, 714, 737, 623, 708, 868, 723, 707, 676, 642, 833, 776

Offset: 0

Labos Elemer, Sep 26 2002

nonn

			n=2: 5^n=25, list={25, 76, 38, 19, 58, 29, 88, 44, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10.5, 16, 8, 4, 2, 1}, a(2)=24.

T. D. Noe, Table of n, a(n) for n = 0..1000

Cf. A006370, A008908, A074472, A075485, A075486, A075487, A075488.

Table[Length[NestWhileList[If[EvenQ@ #, #/2, 3 # + 1] &, 5^n, # != 1 &]], {n, 0, 52}] (* Michael De Vlieger, Feb 25 2017 *)

a(n) = A008908(5^n).

A075486 Length of iteration list when Collatz-function is iterated with initial value 2^n + 1.

Original entry on oeis.org

8, 6, 20, 13, 27, 28, 122, 123, 36, 37, 157, 114, 53, 54, 99, 100, 101, 102, 103, 73, 167, 168, 169, 170, 171, 172, 248, 174, 188, 189, 252, 253, 179, 180, 318, 244, 196, 197, 154, 155, 156, 157, 401, 327, 496, 497, 162, 163, 332, 333, 409, 472, 411, 412, 338

Offset: 1

Labos Elemer, Sep 26 2002

nonn

n=4, 1+2^n = 17, list = {17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1}, so a(4) = 13.

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

Cf. A006370, A008908, A074472, A075487-A075488.

Collatz[n_] := NestWhileList[If[EvenQ[#], #/2, 3 # + 1] &, n, # > 1 &]; Table[Length[Collatz[2^n + 1]], {n, 100}] (* T. D. Noe, Jan 17 2013 *)

a(n) = A008908(2^n+1).

A179118 Number of Collatz steps to reach 1 starting with 2^n + 1.

Original entry on oeis.org

1, 7, 5, 19, 12, 26, 27, 121, 122, 35, 36, 156, 113, 52, 53, 98, 99, 100, 101, 102, 72, 166, 167, 168, 169, 170, 171, 247, 173, 187, 188, 251, 252, 178, 179, 317, 243, 195, 196, 153, 154, 155, 156, 400, 326, 495, 496, 161, 162, 331, 332, 408, 471, 410, 411, 337, 338, 339, 340, 553

Offset: 0

Mitch Harris, Jan 04 2011

There are many long runs of consecutive terms that increase by 1 (see second conjecture in A277109). For n < 40000, the longest run has 1030 terms starting from a(33237) = 244868 and ending with a(34266) = 245897. - Dmitry Kamenetsky, Sep 30 2016

			a(1)=7 because the trajectory of 2^1+1=3 is (3,10,5,16,8,4,2,1).

Dmitry Kamenetsky, Table of n, a(n) for n = 0..9999 (terms n = 1...1000 from Mitch Harris)
MathOverflow, Introduced on Mathoverflow by Henrik Rüping

Cf. A000051, A006577, A070976, A074472, A075486, A193688 (starting with 2^n-1), , A179118, A277109.

CollatzNext[n_] := If[Mod[n, 2] == 0, n/2, 3 n + 1]; CollatzPath[n_] := CollatzPath[n] = Module[{k = n, l = {}}, While[k != 1, k = CollatzNext[k]; l = Append[l, k]]; l]; Collatz[n_] := Length[CollatzPath[n]]; Table[Collatz[2^n+1],{n,1,50}]
f[n_] := Length@ NestWhileList[If[OddQ@ #, 3 # + 1, #/2] &, 2^n + 1, # > 1 &] - 1; Array[f, 60] (* Robert G. Wilson v, Jan 05 2011 *)
Array[-1 + Length@ NestWhileList[If[EvenQ@ #, #/2, 3 # + 1] &, 2^# + 1, # > 1 &] &, 60, 0] (* Michael De Vlieger, Nov 25 2018 *)

nbsteps(n)= s=n; c=0; while(s>1, s=if(s%2, 3*s+1, s/2); c++); c;
a(n) = nbsteps(2^n+1); \\ Michel Marcus, Oct 28 2018

def steps(a):
  if a==1:     return 0
  elif a%2==0: return 1+steps(a//2)
  else:        return 1+steps(a*3+1)
for n in range(60):
  print(n, steps((1<

a(n) = A006577(2^n+1) = A006577(A000051(n)).

a(n) = A075486(n) - 1. - T. D. Noe, Jan 17 2013

a(0)=1 prepended by Alois P. Heinz, Dec 12 2018

A070976 Number of steps to reach 1 in '3x+1' (or Collatz) problem starting with 3^n.

Original entry on oeis.org

0, 7, 19, 111, 22, 96, 33, 76, 75, 43, 135, 134, 133, 132, 144, 205, 129, 190, 140, 95, 94, 261, 428, 91, 258, 394, 331, 255, 254, 390, 389, 388, 462, 461, 460, 459, 458, 457, 456, 455, 454, 453, 501, 500, 450, 498, 497, 752, 495, 494, 493, 748, 491, 746, 489

Offset: 0

Benoit Cloitre, May 17 2002

For all n, it appears that a(n) <= 37n. For n > 22, it appears that a(n) < 16n. - T. D. Noe, Feb 02 2007

This sequence contains some unusually long runs of values that differ by 1. - Dmitry Kamenetsky, Dec 09 2016

			For n=4, 3^4 = 81, and the Collatz sequence (3x + 1 if odd, x/2 if even) goes 81, 244, 122, 61, 184, 92, 46, 23, 70, 35, 106, 53, 160, 80, 40, 20, 10, 5, 16, 8, 4, 2, 1.  This is 22 steps, so a(4) = 22. - _Michael B. Porter_, Dec 15 2016

T. D. Noe, Table of n, a(n) for n = 0..2000

Cf. A279269, A277109. Equals A006577(3^n).

Table[Length[NestWhileList[If[EvenQ[#],#/2,3#+1]&,3^n,#>1&]]-1, {n,0,60}] (* Harvey P. Dale, Oct 13 2011 *)

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

a(n) = A075487(n+1) = A074472(n) + 1. - T. D. Noe, Feb 02 2007

A075485 Length of iteration list when Collatz-function is iterated with initial value 2^n - 1.

Original entry on oeis.org

1, 8, 17, 18, 107, 108, 47, 48, 62, 63, 157, 158, 159, 160, 130, 131, 225, 226, 178, 179, 304, 305, 474, 475, 445, 446, 385, 386, 449, 450, 451, 452, 528, 529, 530, 531, 532, 533, 534, 535, 536, 537, 587, 588, 589, 590, 591, 592, 593, 594, 595, 596, 853, 854

Offset: 1

Labos Elemer, Sep 26 2002

nonn

Somewhat surprisingly, these iterations take almost twice as long as the iterations for 2^n + 1. See A075486. - T. D. Noe, Jan 17 2013

			n=4, 2^n - 1 = 15, list = {15, 46, 23, 70, 35, 106, 53, 160, 80, 40, 20, 10, 5, 16, 8, 4, 2, 1}, so a(4) = 18.

Cf. A006370, A008908, A074472, A075484, A075486, A075487, A075488.

Collatz[n_] := NestWhileList[If[EvenQ[#], #/2, 3 # + 1] &, n, # > 1 &]; Table[Length[Collatz[2^n - 1]], {n, 100}] (* T. D. Noe, Jan 17 2013 *)

a(n) = A008908(2^n-1).

cp's OEIS Frontend

A075487 Length of iteration list when Collatz-function is iterated with initial value 1+3^n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A074474 Number of iteration that first becomes smaller than the initial value if Collatz-function (A006370) is iterated, starting with numbers of form 4n+3.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Formula

A075483 Number of iteration that first becomes smaller than the initial value if Collatz-function (A006370) is iterated, starting with numbers of form 64n+63.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A075488 Length of iteration list when Collatz-function is iterated with initial value -1+3^n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A075482 Number of iteration that first becomes smaller than the initial value if Collatz-function (A006370) is iterated, starting with numbers of the form 64n + 59.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A075484 Length of iteration-list when Collatz-function(A006370) is iterated with initial value 5^n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A075486 Length of iteration list when Collatz-function is iterated with initial value 2^n + 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A179118 Number of Collatz steps to reach 1 starting with 2^n + 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs