A363270 The result, starting from n, of Collatz steps x -> (3x+1)/2 while odd, followed by x -> x/2 while even.

1, 1, 1, 1, 1, 3, 13, 1, 7, 5, 13, 3, 5, 7, 5, 1, 13, 9, 11, 5, 1, 11, 5, 3, 19, 13, 31, 7, 11, 15, 121, 1, 25, 17, 5, 9, 7, 19, 67, 5, 31, 21, 49, 11, 17, 23, 121, 3, 37, 25, 29, 13, 5, 27, 47, 7, 43, 29, 67, 15, 23, 31, 91, 1, 49, 33, 19, 17, 13, 35, 121, 9, 55

Offset: 1

Author

Dustin Theriault, May 23 2023

Keywords

• nonn
• easy

Comments

Each x -> (3x+1)/2 step decreases the number of trailing 1-bits by 1 so A007814(n+1) of them, and the result of those steps is 2*A085062(n).

Links

• Dustin Theriault, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000265, A085062.

Cf. A160541 (number of iterations).

Cf. A075677.

Programs

C

int a(int n) {
  while (n & 1) n += (n >> 1) + 1;
  while (!(n & 1)) n >>= 1;
  return n;
}

Mathematica

OddPart[x_] := x / 2^IntegerExponent[x, 2]
Table[OddPart[(3/2)^IntegerExponent[i + 1, 2] * (i + 1) - 1], {i, 100}]

PARI

oddpart(n) = n >> valuation(n, 2); \\ A000265
a(n) = oddpart((3/2)^valuation(n+1, 2)*(n+1) - 1); \\ Michel Marcus, May 24 2023

Formula

a(n) = OddPart((3/2)^A007814(n+1)*(n+1) - 1), where OddPart(t) = A000265(t).

a(n) = OddPart(A085062(n)).