A273126 Length of "continued logarithm" expansion of n.

1, 1, 2, 1, 2, 2, 4, 1, 2, 2, 4, 2, 5, 4, 6, 1, 2, 2, 4, 2, 5, 4, 6, 2, 7, 5, 5, 4, 5, 6, 8, 1, 2, 2, 4, 2, 5, 4, 6, 2, 7, 5, 5, 4, 5, 6, 8, 2, 9, 7, 6, 5, 6, 5, 6, 4, 8, 5, 7, 6, 6, 8, 10, 1, 2, 2, 4, 2, 5, 4, 6, 2, 7, 5, 5, 4, 5, 6, 8, 2, 9, 7, 6, 5, 6, 5, 6, 4, 8, 5, 7, 6, 6, 8, 10, 2, 11, 9, 7, 7, 8, 6, 9, 5, 7, 6, 8, 5, 7, 6, 9, 4, 9, 8, 11, 5, 6, 7, 7, 6, 7, 6, 8, 8, 9, 10, 12

Offset

1,3

Comments

It is known that a(n) is bounded by 2 log_2 (n) + 2; see my preprint linked below. - Jeffrey Shallit, Jun 14 2016

Links

Rémy Sigrist, Table of n, a(n) for n = 1..16384

Jon Borwein, Neil Calkin, Scott Lindstrom, and Andrew Mattingly, Continued logarithms and associated continued fractions, preprint, 2016.

J. Shallit, Length of the continued logarithm algorithm on rational inputs, arXiv:1606.03881 [math.NT], June 13 2016.

Example

The expansions for n=2 to 19 are [1], [1,1], [2], [2,2], [2,1], [2,0,1,1], [3], [3,3], [3,2], [3,1,1,1], [3,1], [3,0,0,0,1], [3,0,1,1], [3,0,2,0,1,1], [4], [4,4], [4,3], [4,2,1,1]. - R. J. Mathar, Jun 02 2016
Displayed as a table with row lengths A000079 as suggested by a(A000079(k)) =1: - R. J. Mathar, Jun 04 2016
1,
1,2,
1,2,2,4,
1,2,2,4,2,5,4,6,
1,2,2,4,2,5,4,6,2,7,5,5,4,5,6,8,
1,2,2,4,2,5,4,6,2,7,5,5,4,5,6,8,2,9,7,6,5,6,5,6,4,8,5,7,6,6,8,10,
1,2,2,4,2,5,4,6,2,7,5,5,4,5,6,8,2,9,7,6,5,6,5,6,4,8,5,7,6,6,8,10,2,11,9,7,7,8,6,9,5,7,6,8,5,7,6,9,4,9,8,11,5,6,7,7,6,7,6,8,8,9,10,12

Maple

A273126 := proc(n)
    local a , x, cf;
    if n = 1 then
        return 1;
    end if;
    cf := [] ;
    x := n ;
    while x > 1 do
        a := ilog2(x) ;
        cf := [op(cf), a] ;
        x := x/2^a ;
        if x = 1 then
            break;
        end if;
        x := 1/(x-1) ;
    end do:
    nops(cf) ;
end proc:
seq(A273126(n), n=1..80) ; # R. J. Mathar, Jun 02 2016

Prog

(PARI) a(n) = my (x=n); for (w=1, oo, while (x>=2, x /= 2); if (x==1, return (w)); x = 1/(x-1); if (x<=1, return (w))) \\ Rémy Sigrist, Sep 09 2018

Crossrefs

Sequence in context: A048896 A130831 A151678 * A151681 A336138 A366600

Adjacent sequences: A273123 A273124 A273125 * A273127 A273128 A273129

Keyword

nonn

Author

Jeffrey Shallit, May 16 2016

Status

approved