login
A273126
Length of "continued logarithm" expansion of n.
2
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
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
KEYWORD
nonn
AUTHOR
Jeffrey Shallit, May 16 2016
STATUS
approved