A341194 -id:A341194 - cp's OEIS frontend

A340873 a(n) is the number of iterations of A245471 needed to reach 1 starting from n.

0, 1, 3, 2, 6, 4, 4, 3, 11, 7, 7, 5, 9, 5, 5, 4, 12, 12, 12, 8, 8, 8, 8, 6, 10, 10, 10, 6, 14, 6, 6, 5, 21, 13, 13, 13, 13, 13, 13, 9, 17, 9, 9, 9, 17, 9, 9, 7, 19, 11, 11, 11, 11, 11, 11, 7, 15, 15, 15, 7, 15, 7, 7, 6, 22, 22, 22, 14, 14, 14, 14, 14, 22, 14

Offset: 1

Rémy Sigrist, Jan 31 2021

This sequence is well defined.
Sketch of proof:
- we focus on odd numbers n > 1,
- if the binary representation of n ends with k 0's and one 1:
       in two steps we obtain a number with the same binary length as n
       and ending with k-1 0's and one 1,
       iterating again will eventually give a number ending with two or more 1's,
- if the binary representation of n ends with k 1's (k > 1):
       in k+1 steps we obtain a number with a binary length strictly smaller
       than that of n,
- so any odd number > 1 will eventually reach the number 1.

			For n = 10:
- the trajectory of 10 is 10 -> 5 -> 14 -> 7 -> 8 -> 4 -> 2 -> 1,
- so a(10) = 7.

Cf. A006577, A245471, A341194, A341218, A341220, A341231, A341235.

a(n) = for (k=0, oo, if (n==1, return (k), n%2, n=bitxor(n, 2*n+1), n=n/2))

a(2*n) = a(n) + 1.

A341220 Irregular table read by rows T(n, k), n >= 0, k = 1..A341218(n); T(n, k) is the k-th number m such that A340873(m) = n.

Original entry on oeis.org

1, 2, 4, 3, 8, 6, 7, 16, 12, 14, 15, 32, 5, 24, 28, 30, 31, 64, 10, 11, 48, 56, 60, 62, 63, 128, 20, 21, 22, 23, 96, 112, 120, 124, 126, 127, 256, 13, 40, 42, 43, 44, 46, 47, 192, 224, 240, 248, 252, 254, 255, 512, 25, 26, 27, 80, 84, 85, 86, 87, 88, 92, 94, 95, 384, 448, 480, 496, 504, 508, 510, 511, 1024

Offset: 0

Rémy Sigrist, Feb 07 2021

Each positive integer appears once.
For any n >= 0:
- if m appears in row n, then 2*m appears in row n+1,
- if m appears in row n and is an odious even number > 2, then an additional odd number appears in row n+1.

			Table begins:
    1;
    2;
    4;
    3, 8;
    6, 7, 16;
    12, 14, 15, 32;
    5, 24, 28, 30, 31, 64;
    10, 11, 48, 56, 60, 62, 63, 128;
    20, 21, 22, 23, 96, 112, 120, 124, 126, 127, 256;
    13, 40, 42, 43, 44, 46, 47, 192, 224, 240, 248, 252, 254, 255, 512;
    ...

Rémy Sigrist, Table of n, a(n) for n = 0..11792 (rows 0..26)
Rémy Sigrist, PARI program for A341220
Index entries for sequences related to 3x+1 (or Collatz) problem
Index entries for sequences that are permutations of the natural numbers

Cf. A127824, A340873, A341194, A341218.

PARI
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See Links section.
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T(n, 1) = A341194(n).

T(n, A341218(n)) = 2^n.

cp's OEIS Frontend

A340873 a(n) is the number of iterations of A245471 needed to reach 1 starting from n.

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