A074474 -id:A074474 - cp's OEIS frontend

A075476 Number of iteration that first becomes smaller than the initial value if Collatz-function (A006370) is iterated, starting with numbers of form 64n+7. Corresponds to selection of every 16th term from A074474.

12, 84, 12, 14, 12, 35, 12, 14, 12, 17, 12, 14, 12, 25, 12, 14, 12, 25, 12, 14, 12, 27, 12, 14, 12, 17, 12, 14, 12, 38, 12, 14, 12, 25, 12, 14, 12, 45, 12, 14, 12, 17, 12, 14, 12, 27, 12, 14, 12, 20, 12, 14, 12, 79, 12, 14, 12, 17, 12, 14, 12, 20, 12, 14, 12, 33, 12, 14, 12

Offset: 0

Labos Elemer, Sep 23 2002

nonn

Remark that initial values of form 64m+r, if r={3, 11, 19, 27, 35, 43, 51, 55} provide first-sink-lengths {7, 9, 7, 9, 7, 9, 7, 9} respectively; e.g. {64k+19, 192k+58, 96k+29, 288k+88, 144k+44, 72k+22, 36k+11} submerge first below initial value at the 7th term, 36k+11<64k+19.

			n=0: 64n+7=7, list={7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5..}, i.e. the 12th term is the first that <12, the initial value.

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

Cf. A074473, A074474, A006370.

lcoll[n_] := Length[NestWhileList[If[EvenQ[#], #/2, 3 # + 1] &, n, # >= n &]]; Table[lcoll[64*i + 7], {i, 0, 68}] (* Jayanta Basu, Jun 15 2013 *)

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)));
A075476(n) = A074473((64*n)+7); \\ Antti Karttunen, Oct 09 2018

a(n) = A074473(64n+7), n=0, ..., 256

Typo in formula corrected by Antti Karttunen, Oct 09 2018

A075477 Number of iteration that first becomes smaller than the initial value if Collatz-function (A006370) is iterated, starting with numbers of form 64n+15. Corresponds to selection of every 16th term from A074474.

Original entry on oeis.org

12, 14, 12, 22, 12, 14, 12, 20, 12, 14, 12, 22, 12, 14, 12, 17, 12, 14, 12, 20, 12, 14, 12, 40, 12, 14, 12, 58, 12, 14, 12, 17, 12, 14, 12, 33, 12, 14, 12, 33, 12, 14, 12, 25, 12, 14, 12, 17, 12, 14, 12, 33, 12, 14, 12, 27, 12, 14, 12, 40, 12, 14, 12, 17, 12, 14, 12, 69, 12

Offset: 0

Labos Elemer, Sep 23 2002

nonn

Remark that initial values of form 64m+r, if r={3,11,19,27,35,43,51,55} provide first-sink-lengths {7,9,7,9,7,9,7,9} respectively; e.g. {64k+19,192k+58,96k+29,288k+88,144k+44,72k+22,36k+11} submerge first below initial value at the 7th term,36k+11<64k+19.

			n=0: 64n+15=15,list={15,46,23,70,35,106,53,160,80,40,20,10..}, i.e. the 12th term is the first that <15, the initial value.

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

Cf. A074473, A074474, A075476-A075482, A006370.

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)));
A075477(n) = A074473((64*n)+15); \\ Antti Karttunen, Oct 09 2018

a(n) = A074473(64n+15), n=0..256. [corrected by Antti Karttunen, Oct 09 2018]

A075478 Number of iteration that first becomes smaller than the initial value if Collatz-function (A006370) is iterated, starting with numbers of form 64n+27. Corresponds to selection of every 16th term from A074474.

Original entry on oeis.org

97, 74, 66, 14, 40, 17, 25, 14, 22, 27, 40, 14, 45, 27, 17, 14, 40, 38, 27, 14, 56, 17, 20, 14, 22, 27, 30, 14, 100, 30, 17, 14, 22, 33, 20, 14, 22, 17, 30, 14, 20, 30, 53, 14, 38, 20, 17, 14, 51, 25, 66, 14, 35, 17, 22, 14, 25, 20, 64, 14, 38, 40, 17, 14, 45, 25, 22, 14, 27

Offset: 0

Labos Elemer, Sep 23 2002

nonn

Remark that initial values of form 64m+r, if r={3,11,19,27,35,43,51,55} provide first-sink-lengths {7,9,7,9,7,9,7,9} respectively; e.g. {64k+19,192k+58,96k+29,288k+88,144k+44,72k+22,36k+11} submerge first below initial value at the 7th term,36k+11<64k+19.

			n=0: 64n+27=27, list={27, 82, 41, 46.23.70, ..}, i.e. the 97th term is the first that <27, the initial value.

Cf. A074473, A074474, A075476-A075482, A006370.

a(n)=A075473[64n+27], n=0, ..., 256

A075479 Number of iteration that first becomes smaller than the initial value if Collatz-function (A006370) is iterated, starting with numbers of form 64n+31. Corresponds to selection of every 16th term from A074474.

Original entry on oeis.org

92, 14, 35, 51, 17, 14, 25, 27, 22, 14, 64, 17, 22, 14, 61, 43, 131, 14, 27, 22, 17, 14, 33, 35, 22, 14, 53, 17, 20, 14, 43, 22, 22, 14, 45, 22, 17, 14, 35, 43, 20, 14, 25, 17, 25, 14, 20, 22, 27, 14, 38, 20, 17, 14, 27, 22, 30, 14, 25, 17, 33, 14, 40, 20, 69, 14, 115, 27, 17

Offset: 0

Labos Elemer, Sep 23 2002

nonn

Remark that initial values of form 64m+r, if r={3,11,19,27,35,43,51,55} provide first-sink-lengths {7,9,7,9,7,9,7,9} respectively; e.g. {64k+19,192k+58,96k+29,288k+88,144k+44,72k+22,36k+11} submerge first below initial value at the 7th term,36k+11<64k+19.

			n=1: 64n+31=95,list={95,286,143,430,215,646,323,970, 485,1456,728,364,182,91,274,...}, i.e. the 14th term=91 is the first that <95, the initial value, so a(1)=14.

Cf. A074473, A074474, A075476-A075482, A006370.

a(n)=A075473[64n+31], n=0, ..., 256

A075481 Number of iteration that first becomes smaller than the initial value if Collatz-function (A006370) is iterated, starting with numbers of form 64n+47. Corresponds to selection of every 16th term from A074474.

Original entry on oeis.org

89, 51, 14, 33, 22, 17, 14, 45, 27, 22, 14, 35, 17, 20, 14, 35, 22, 22, 14, 43, 22, 17, 14, 27, 128, 20, 14, 25, 17, 25, 14, 20, 22, 30, 14, 82, 20, 17, 14, 45, 22, 27, 14, 25, 17, 27, 14, 48, 20, 30, 14, 43, 30, 17, 14, 58, 61, 27, 14, 53, 17, 56, 14, 22, 30, 58, 14, 27, 53

Offset: 0

Labos Elemer, Sep 23 2002

nonn

Remark that initial values of form 64m+r, if r={3,11,19,27,35,43,51,55} provide first-sink-lengths {7,9,7,9,7,9,7,9} respectively; e.g. {64k+19,192k+58,96k+29,288k+88,144k+44,72k+22,36k+11} submerge first below initial value at the 7th term,36k+11<64k+19.

			n=2: 64n+47=175,list={175,526,263,790,395,1186,593,1780, 890,445,1336,668,334,167,502,251....}, i.e. the 14th term=167 is the first that <175, the initial value, so a(2)=14.

Cf. A074473, A074474, A075476-A075483, A006370.

a(n)=A075473[64n+47], n=0, ..., 256

A075480 Number of iteration that first becomes smaller than the initial value if Collatz function (A006370) is iterated, starting with numbers of the form 64n + 39.

Original entry on oeis.org

14, 69, 48, 20, 14, 27, 17, 33, 14, 20, 22, 40, 14, 58, 20, 17, 14, 33, 22, 33, 14, 64, 17, 33, 14, 71, 20, 35, 14, 40, 43, 17, 14, 71, 71, 25, 14, 27, 17, 40, 14, 22, 25, 27, 14, 43, 25, 17, 14, 66, 27, 25, 14, 76, 17, 20, 14, 22, 43, 27, 14, 66, 25, 17, 14, 22

Offset: 0

Labos Elemer, Sep 23 2002

Initial values of the form 64m + r, if r = {3,11,19,27,35,43,51,55}, provide first-sink-lengths {7,9,7,9,7,9,7,9} respectively; e.g., {64k + 19, 192k + 58, 96k + 29, 288k + 88, 144k + 44, 72k + 22, 36k + 11} submerge first below initial value at the 7th term, 36k + 11 < 64k + 19.

			n=0: 64n + 39 = 39, Collatz trajectory = {39, 118, 59, 178, 89, 268, 134, 67, 202, 101, 304, 152, 76, 38, 19, 58, ....}, i.e., the 14th term = 38 is the first that is less than 39, the initial value, so a(0)=14.

Nathaniel Johnston, Table of n, a(n) for n = 0..10000

Cf. A074473, A074474, A075476-A075483, A006370.

col := proc(n) if(n mod 2 = 0)then return n/2: fi: return 3*n+1: end: A075480 := proc(n) local s,v: s:=1: v:=64*n+39: while v>=64*n+39 do v:=col(v): s:=s+1: od: return s: end: seq(A075480(n),n=0..65); # Nathaniel Johnston, Jun 22 2011

a(n) = A074473(64n+39).

Keyword:fini removed by Nathaniel Johnston, Jun 23 2011

Edited by Jon E. Schoenfield, Feb 23 2019

A361733 Length of the Collatz (3x + 1) trajectory from k = 10^n - 1 to a term less than k, or -1 if the trajectory never goes below k.

Original entry on oeis.org

4, 7, 17, 12, 113, 17, 79, 22, 51, 33, 64, 35, 128, 56, 110, 53, 84, 128, 107, 115, 175, 82, 477, 172, 141, 182, 188, 110, 159, 167, 301, 206, 151, 146, 128, 195, 190, 299, 208, 276, 180, 185, 500, 203, 229, 190, 265, 270, 288, 252, 299, 208, 350, 348, 459, 330, 314, 268, 490, 361, 578

Offset: 1

Paul M. Bradley, Mar 22 2023

nonn

k = 10^n - 1 = A002283(n) is the repdigit consisting of n digits, all 9s.
The sequence seems to be chaotic but broadly increasing.
By contrast, repdigits of 1, 3, 5, or 7, have constant dropping times after a few initial values each.

			a(1) = 4 as for k = 9, the Collatz trajectory begins 9, 28, 14, 7, ...;
a(2) = 7 as for k = 99, the Collatz trajectory begins 99, 298, 149, 448, 224, 112, 56, ...;
a(3) = 17 as for k = 999, the Collatz trajectory begins 999, 2998, 1499, 4498, 2249, 6748, 3374, 1687, 5062, 2531, 7594, 3797, 11392, 5696, 2848, 1424, 712, ... .

Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A074473, A002283, A006370, A074474, A075483, A075480.

collatzLen[a_Integer] := Module[{len = 1, x = a},
  While[x >= a,    If[Mod[x, 2] > 0,
      x = 3 x + 1,
      x = Quotient[x, 2]
    ];
    len++
  ];
  Return[len]
]

f(n) = if (n%2, 3*n+1, n/2); \\ A006370
b(n) = if (n<3, return(n)); my(m=n, nb=0); while (1, m = f(m); nb++; if (m < n, return(nb+1));); \\ A074473
a(n) = b(10^n-1); \\ Michel Marcus, Mar 28 2023

def collatz_len(a):
    length = 1
    x = a
    while x >= a:
        if x % 2 > 0:
            x = 3 * x + 1
        else:
            x = x // 2
        length += 1
    return length

a(n) = A074473(10^n-1).

cp's OEIS Frontend

A075476 Number of iteration that first becomes smaller than the initial value if Collatz-function (A006370) is iterated, starting with numbers of form 64n+7. Corresponds to selection of every 16th term from A074474.

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A075477 Number of iteration that first becomes smaller than the initial value if Collatz-function (A006370) is iterated, starting with numbers of form 64n+15. Corresponds to selection of every 16th term from A074474.

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A075478 Number of iteration that first becomes smaller than the initial value if Collatz-function (A006370) is iterated, starting with numbers of form 64n+27. Corresponds to selection of every 16th term from A074474.

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A075479 Number of iteration that first becomes smaller than the initial value if Collatz-function (A006370) is iterated, starting with numbers of form 64n+31. Corresponds to selection of every 16th term from A074474.

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A075481 Number of iteration that first becomes smaller than the initial value if Collatz-function (A006370) is iterated, starting with numbers of form 64n+47. Corresponds to selection of every 16th term from A074474.

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A075480 Number of iteration that first becomes smaller than the initial value if Collatz function (A006370) is iterated, starting with numbers of the form 64n + 39.

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A361733 Length of the Collatz (3x + 1) trajectory from k = 10^n - 1 to a term less than k, or -1 if the trajectory never goes below k.

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