A341235 -id:A341235 - 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.

A341231 Irregular triangle read by rows giving trajectory from n to reach 1 under the map A245471.

Original entry on oeis.org

1, 2, 1, 3, 4, 2, 1, 4, 2, 1, 5, 14, 7, 8, 4, 2, 1, 6, 3, 4, 2, 1, 7, 8, 4, 2, 1, 8, 4, 2, 1, 9, 26, 13, 22, 11, 28, 14, 7, 8, 4, 2, 1, 10, 5, 14, 7, 8, 4, 2, 1, 11, 28, 14, 7, 8, 4, 2, 1, 12, 6, 3, 4, 2, 1, 13, 22, 11, 28, 14, 7, 8, 4, 2, 1, 14, 7, 8, 4, 2, 1

Offset: 1

Rémy Sigrist, Feb 07 2021

A340873 gives row lengths.

A341235 gives greatest terms.

			Table begins:
    1;
    2, 1;
    3, 4, 2, 1;
    4, 2, 1;
    5, 14, 7, 8, 4, 2, 1;
    6, 3, 4, 2, 1;
    7, 8, 4, 2, 1;
    8, 4, 2, 1;
    9, 26, 13, 22, 11, 28, 14, 7, 8, 4, 2, 1;
    10, 5, 14, 7, 8, 4, 2, 1;
    11, 28, 14, 7, 8, 4, 2, 1;
    12, 6, 3, 4, 2, 1;
    13, 22, 11, 28, 14, 7, 8, 4, 2, 1;
    14, 7, 8, 4, 2, 1;
    15, 16, 8, 4, 2, 1;
    16, 8, 4, 2, 1;
    ...

Rémy Sigrist, Table of n, a(n) for n = 1..9098 (rows 1..500)

Cf. A070165, A245471, A340873, A341235.

row(n) = { my (r=[n]); while (n>1, r=concat(r, n=if (n%2, bitxor(n, 2*n+1), n/2))); r }

T(n, 1) = n.

T(n, A340873(n)) = 1.

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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