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.

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

1,1

Comments

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.

Links

Table of n, a(n) for n=1..61.

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

Formula

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

Example

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, ... .

Mathematica

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]
]

Prog

(Python)
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
(PARI) 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

Crossrefs

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

Sequence in context: A340600 A013625 A182929 * A367744 A363642 A124402

Adjacent sequences: A361730 A361731 A361732 * A361734 A361735 A361736

Keyword

nonn

Author

Paul M. Bradley, Mar 22 2023

Status

approved