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

Author

Paul M. Bradley, Mar 22 2023

Keywords

• nonn

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.

Examples

			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, ... .
		

Links

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

Crossrefs

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

Programs

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]
]

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

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

Formula

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